Properties

Label 8-48e4-1.1-c16e4-0-2
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $3.68554\times 10^{7}$
Root an. cond. $8.82699$
Motivic weight $16$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.05e3·3-s + 3.14e6·7-s + 1.14e7·9-s − 1.58e9·13-s + 5.61e10·19-s + 6.44e9·21-s + 3.01e11·25-s + 1.26e11·27-s − 2.47e12·31-s + 3.70e11·37-s − 3.24e12·39-s + 2.80e13·43-s − 4.67e13·49-s + 1.15e14·57-s + 3.62e14·61-s + 3.58e13·63-s + 2.67e13·67-s + 3.17e14·73-s + 6.17e14·75-s − 4.79e15·79-s − 1.32e15·81-s − 4.96e15·91-s − 5.07e15·93-s − 1.38e16·97-s + 2.29e16·103-s + 2.12e16·109-s + 7.60e14·111-s + ⋯
L(s)  = 1  + 0.312·3-s + 0.544·7-s + 0.265·9-s − 1.93·13-s + 3.30·19-s + 0.170·21-s + 1.97·25-s + 0.448·27-s − 2.89·31-s + 0.105·37-s − 0.606·39-s + 2.40·43-s − 1.40·49-s + 1.03·57-s + 1.88·61-s + 0.144·63-s + 0.0659·67-s + 0.393·73-s + 0.617·75-s − 3.15·79-s − 0.717·81-s − 1.05·91-s − 0.906·93-s − 1.76·97-s + 1.81·103-s + 1.06·109-s + 0.0329·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+8)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(3.68554\times 10^{7}\)
Root analytic conductor: \(8.82699\)
Motivic weight: \(16\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :8, 8, 8, 8),\ 1)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(12.51909447\)
\(L(\frac12)\) \(\approx\) \(12.51909447\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$D_{4}$ \( 1 - 76 p^{3} T - 122 p^{10} T^{2} - 76 p^{19} T^{3} + p^{32} T^{4} \)
good5$C_2^2 \wr C_2$ \( 1 - 12045550276 p^{2} T^{2} + 20208485552534687982 p^{5} T^{4} - 12045550276 p^{34} T^{6} + p^{64} T^{8} \)
7$D_{4}$ \( ( 1 - 224396 p T + 79000079514 p^{3} T^{2} - 224396 p^{17} T^{3} + p^{32} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 - 5004710841163244 p T^{2} + \)\(15\!\cdots\!26\)\( p^{3} T^{4} - 5004710841163244 p^{33} T^{6} + p^{64} T^{8} \)
13$D_{4}$ \( ( 1 + 60797324 p T + 8193950692930518 p^{2} T^{2} + 60797324 p^{17} T^{3} + p^{32} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 - \)\(13\!\cdots\!60\)\( T^{2} + \)\(81\!\cdots\!58\)\( T^{4} - \)\(13\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \)
19$D_{4}$ \( ( 1 - 1476766220 p T + \)\(42\!\cdots\!98\)\( T^{2} - 1476766220 p^{17} T^{3} + p^{32} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - \)\(90\!\cdots\!80\)\( p T^{2} + \)\(18\!\cdots\!78\)\( T^{4} - \)\(90\!\cdots\!80\)\( p^{33} T^{6} + p^{64} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - \)\(48\!\cdots\!44\)\( T^{2} + \)\(17\!\cdots\!66\)\( T^{4} - \)\(48\!\cdots\!44\)\( p^{32} T^{6} + p^{64} T^{8} \)
31$D_{4}$ \( ( 1 + 39867438004 p T + \)\(18\!\cdots\!06\)\( T^{2} + 39867438004 p^{17} T^{3} + p^{32} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 185281606948 T + \)\(24\!\cdots\!22\)\( T^{2} - 185281606948 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - \)\(23\!\cdots\!04\)\( T^{2} + \)\(21\!\cdots\!66\)\( T^{4} - \)\(23\!\cdots\!04\)\( p^{32} T^{6} + p^{64} T^{8} \)
43$D_{4}$ \( ( 1 - 14032511031332 T + \)\(22\!\cdots\!02\)\( T^{2} - 14032511031332 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - \)\(18\!\cdots\!60\)\( T^{2} + \)\(14\!\cdots\!98\)\( T^{4} - \)\(18\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!60\)\( T^{2} + \)\(71\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - \)\(16\!\cdots\!04\)\( T^{2} + \)\(93\!\cdots\!66\)\( T^{4} - \)\(16\!\cdots\!04\)\( p^{32} T^{6} + p^{64} T^{8} \)
61$D_{4}$ \( ( 1 - 181134896541604 T + \)\(81\!\cdots\!26\)\( T^{2} - 181134896541604 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 13387202681732 T + \)\(30\!\cdots\!42\)\( T^{2} - 13387202681732 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!44\)\( T^{2} + \)\(66\!\cdots\!66\)\( T^{4} - \)\(12\!\cdots\!44\)\( p^{32} T^{6} + p^{64} T^{8} \)
73$D_{4}$ \( ( 1 - 158814443769988 T + \)\(37\!\cdots\!62\)\( T^{2} - 158814443769988 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 2397008582592460 T + \)\(59\!\cdots\!22\)\( p T^{2} + 2397008582592460 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(46\!\cdots\!60\)\( T^{2} - \)\(18\!\cdots\!42\)\( T^{4} - \)\(46\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - \)\(45\!\cdots\!84\)\( T^{2} + \)\(22\!\cdots\!06\)\( T^{4} - \)\(45\!\cdots\!84\)\( p^{32} T^{6} + p^{64} T^{8} \)
97$D_{4}$ \( ( 1 + 6916897501300892 T + \)\(13\!\cdots\!62\)\( T^{2} + 6916897501300892 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.404160607166238453502876995491, −7.915199454407847388379274160287, −7.30822469165074923662167706425, −7.30647881628219163104205791174, −7.27583932508569940966432771542, −7.15321727119795502138509747737, −6.43050333918796660000800849953, −5.94566701661500001534469357456, −5.66297741931799906818096686219, −5.16637380752656585090155679284, −5.07600713195087943887295100507, −5.06336646668085243041877501222, −4.33541372403533142159398466855, −4.18645336544753540424411492244, −3.62425753045779483008799071472, −3.36808698482186719815769844858, −2.83480920281083552414792781319, −2.72275489819538718014157433696, −2.60144086744432061231989706238, −1.75594527123529137069806941134, −1.55006538286977855589162415787, −1.54365727141914777179950297510, −0.66258488982524236975550899194, −0.56713079211566111547128188704, −0.53412304477369861903718761644, 0.53412304477369861903718761644, 0.56713079211566111547128188704, 0.66258488982524236975550899194, 1.54365727141914777179950297510, 1.55006538286977855589162415787, 1.75594527123529137069806941134, 2.60144086744432061231989706238, 2.72275489819538718014157433696, 2.83480920281083552414792781319, 3.36808698482186719815769844858, 3.62425753045779483008799071472, 4.18645336544753540424411492244, 4.33541372403533142159398466855, 5.06336646668085243041877501222, 5.07600713195087943887295100507, 5.16637380752656585090155679284, 5.66297741931799906818096686219, 5.94566701661500001534469357456, 6.43050333918796660000800849953, 7.15321727119795502138509747737, 7.27583932508569940966432771542, 7.30647881628219163104205791174, 7.30822469165074923662167706425, 7.915199454407847388379274160287, 8.404160607166238453502876995491

Graph of the $Z$-function along the critical line