Properties

Label 4-8280e2-1.1-c1e2-0-0
Degree $4$
Conductor $68558400$
Sign $1$
Analytic cond. $4371.34$
Root an. cond. $8.13118$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s − 8·11-s − 2·13-s − 5·17-s + 8·19-s − 2·23-s + 3·25-s + 5·29-s + 31-s − 2·35-s − 5·37-s − 41-s − 8·43-s + 2·47-s − 5·49-s + 3·53-s + 16·55-s − 23·59-s − 4·61-s + 4·65-s + 25·67-s − 71-s − 14·73-s − 8·77-s − 4·79-s − 3·83-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 2.41·11-s − 0.554·13-s − 1.21·17-s + 1.83·19-s − 0.417·23-s + 3/5·25-s + 0.928·29-s + 0.179·31-s − 0.338·35-s − 0.821·37-s − 0.156·41-s − 1.21·43-s + 0.291·47-s − 5/7·49-s + 0.412·53-s + 2.15·55-s − 2.99·59-s − 0.512·61-s + 0.496·65-s + 3.05·67-s − 0.118·71-s − 1.63·73-s − 0.911·77-s − 0.450·79-s − 0.329·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(68558400\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(4371.34\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 68558400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.007041969\)
\(L(\frac12)\) \(\approx\) \(1.007041969\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 5 T + 56 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 5 T + 72 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 23 T + 242 T^{2} + 23 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 25 T + 282 T^{2} - 25 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T + 134 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T + 146 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−7.79922287046638019129428262320, −7.78528995257728834276143300479, −7.33805938904478591516676571468, −7.13185210646970560630039252648, −6.63659154652459136049439924999, −6.39305431292893731603544899779, −5.69536524381127841629776588179, −5.52096834867068834183701973024, −5.12167129017756136471920067878, −4.84616735157180718890589725613, −4.47476889922948518254120293156, −4.34407450425872949378822075948, −3.50699099925323460065889132064, −3.23733522493577617190370047874, −2.88501937440715585415500182619, −2.64064791865709465665266864340, −1.88060025084583719953162132411, −1.73972496766131965581097556829, −0.73272593811362229314989823617, −0.31743108799320733658050039418, 0.31743108799320733658050039418, 0.73272593811362229314989823617, 1.73972496766131965581097556829, 1.88060025084583719953162132411, 2.64064791865709465665266864340, 2.88501937440715585415500182619, 3.23733522493577617190370047874, 3.50699099925323460065889132064, 4.34407450425872949378822075948, 4.47476889922948518254120293156, 4.84616735157180718890589725613, 5.12167129017756136471920067878, 5.52096834867068834183701973024, 5.69536524381127841629776588179, 6.39305431292893731603544899779, 6.63659154652459136049439924999, 7.13185210646970560630039252648, 7.33805938904478591516676571468, 7.78528995257728834276143300479, 7.79922287046638019129428262320

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