Properties

Degree $4$
Conductor $7001316$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 3·5-s + 4·8-s − 6·10-s − 6·11-s − 2·13-s + 5·16-s + 6·17-s + 7·19-s − 9·20-s − 12·22-s + 3·23-s + 5·25-s − 4·26-s + 6·29-s + 4·31-s + 6·32-s + 12·34-s − 2·37-s + 14·38-s − 12·40-s − 2·43-s − 18·44-s + 6·46-s + 10·50-s − 6·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.34·5-s + 1.41·8-s − 1.89·10-s − 1.80·11-s − 0.554·13-s + 5/4·16-s + 1.45·17-s + 1.60·19-s − 2.01·20-s − 2.55·22-s + 0.625·23-s + 25-s − 0.784·26-s + 1.11·29-s + 0.718·31-s + 1.06·32-s + 2.05·34-s − 0.328·37-s + 2.27·38-s − 1.89·40-s − 0.304·43-s − 2.71·44-s + 0.884·46-s + 1.41·50-s − 0.832·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7001316\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2646} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7001316,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.049478911\)
\(L(\frac12)\) \(\approx\) \(5.049478911\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−8.858950556923864377819119323278, −8.482083733865472863739072583068, −8.042890366432348679201748384642, −7.81483241923227660534897359869, −7.38142009381650076942374024511, −7.34813278557772562998255303352, −6.74491255579741140628666536701, −6.37983515279038426289165636690, −5.66977308422121014614917719817, −5.45803972849861360057363719879, −4.97916902072445454831474191262, −4.90697133009652670068329192723, −4.41042664073731870777256956837, −3.69505231734168906113720480524, −3.41390121704826515959585786245, −3.15387774915306592228537812419, −2.56211467694364357154000062474, −2.27440605358588165145481548380, −1.14394034643935806918428788045, −0.65171701235582803956063941108, 0.65171701235582803956063941108, 1.14394034643935806918428788045, 2.27440605358588165145481548380, 2.56211467694364357154000062474, 3.15387774915306592228537812419, 3.41390121704826515959585786245, 3.69505231734168906113720480524, 4.41042664073731870777256956837, 4.90697133009652670068329192723, 4.97916902072445454831474191262, 5.45803972849861360057363719879, 5.66977308422121014614917719817, 6.37983515279038426289165636690, 6.74491255579741140628666536701, 7.34813278557772562998255303352, 7.38142009381650076942374024511, 7.81483241923227660534897359869, 8.042890366432348679201748384642, 8.482083733865472863739072583068, 8.858950556923864377819119323278

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