Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 11 \cdot 37^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s + 2·5-s − 6-s + 7-s − 3·8-s + 9-s + 2·10-s − 11-s + 12-s − 6·13-s + 14-s − 2·15-s − 16-s − 2·17-s + 18-s − 4·19-s − 2·20-s − 21-s − 22-s + 3·24-s − 25-s − 6·26-s − 27-s − 28-s + 2·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.218·21-s − 0.213·22-s + 0.612·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 316239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 316239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(316239\)    =    \(3 \cdot 7 \cdot 11 \cdot 37^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{316239} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 316239,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11,\;37\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11,\;37\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
37 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.02727857861325, −12.75288705026556, −12.36095398663316, −11.77001938770063, −11.43994202048215, −10.76988263052874, −10.39607102489167, −9.892628266872238, −9.487534961802575, −9.088533041591043, −8.612475728456062, −7.925253585969247, −7.493656251501017, −6.897401999037874, −6.443824856470489, −5.830599637708029, −5.509107868950771, −5.147746068154112, −4.547171054930137, −4.259959835544204, −3.702917411320753, −2.696725852767025, −2.526190309725034, −1.865546751780314, −1.142081196647833, 0, 0, 1.142081196647833, 1.865546751780314, 2.526190309725034, 2.696725852767025, 3.702917411320753, 4.259959835544204, 4.547171054930137, 5.147746068154112, 5.509107868950771, 5.830599637708029, 6.443824856470489, 6.897401999037874, 7.493656251501017, 7.925253585969247, 8.612475728456062, 9.088533041591043, 9.487534961802575, 9.892628266872238, 10.39607102489167, 10.76988263052874, 11.43994202048215, 11.77001938770063, 12.36095398663316, 12.75288705026556, 13.02727857861325

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