Properties

Degree $4$
Conductor $88510464$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·5-s + 3·9-s − 4·11-s + 8·15-s + 4·17-s + 8·19-s + 4·23-s + 4·25-s + 4·27-s − 16·31-s − 8·33-s + 8·37-s + 4·41-s + 16·43-s + 12·45-s − 8·47-s + 8·51-s + 4·53-s − 16·55-s + 16·57-s + 16·59-s + 8·61-s + 16·67-s + 8·69-s + 4·71-s + 8·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 9-s − 1.20·11-s + 2.06·15-s + 0.970·17-s + 1.83·19-s + 0.834·23-s + 4/5·25-s + 0.769·27-s − 2.87·31-s − 1.39·33-s + 1.31·37-s + 0.624·41-s + 2.43·43-s + 1.78·45-s − 1.16·47-s + 1.12·51-s + 0.549·53-s − 2.15·55-s + 2.11·57-s + 2.08·59-s + 1.02·61-s + 1.95·67-s + 0.963·69-s + 0.474·71-s + 0.936·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(88510464\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{9408} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(10.61108486\)
\(L(\frac12)\) \(\approx\) \(10.61108486\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 16 T + 118 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 112 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.76535444669679082441199973285, −7.54750418258806786057322371429, −7.22672092762426817910029364460, −7.13191183770128123403107569857, −6.44939456719388754140485567190, −6.14316828410028836453823281925, −5.65193519067449950241227579217, −5.42374577457407386853969695235, −5.16910165833190938267959287824, −5.16507103612409991455515420630, −4.21363644611502808354321523125, −3.94282056281702157547919613250, −3.55736141822403396400456655503, −3.11713948337185822678921272673, −2.66197967867243150344768125470, −2.52338041084155929708728750434, −1.88663731740621090041701346743, −1.84391093301018242056777715998, −0.940470475587217812319300795726, −0.78078301605487281619258244121, 0.78078301605487281619258244121, 0.940470475587217812319300795726, 1.84391093301018242056777715998, 1.88663731740621090041701346743, 2.52338041084155929708728750434, 2.66197967867243150344768125470, 3.11713948337185822678921272673, 3.55736141822403396400456655503, 3.94282056281702157547919613250, 4.21363644611502808354321523125, 5.16507103612409991455515420630, 5.16910165833190938267959287824, 5.42374577457407386853969695235, 5.65193519067449950241227579217, 6.14316828410028836453823281925, 6.44939456719388754140485567190, 7.13191183770128123403107569857, 7.22672092762426817910029364460, 7.54750418258806786057322371429, 7.76535444669679082441199973285

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