Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 7^{2} \cdot 617 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s + 4-s − 5-s + 3·6-s + 8-s + 6·9-s − 10-s − 2·11-s + 3·12-s + 7·13-s − 3·15-s + 16-s − 4·17-s + 6·18-s + 19-s − 20-s − 2·22-s + 3·23-s + 3·24-s + 25-s + 7·26-s + 9·27-s + 9·29-s − 3·30-s − 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s + 0.353·8-s + 2·9-s − 0.316·10-s − 0.603·11-s + 0.866·12-s + 1.94·13-s − 0.774·15-s + 1/4·16-s − 0.970·17-s + 1.41·18-s + 0.229·19-s − 0.223·20-s − 0.426·22-s + 0.625·23-s + 0.612·24-s + 1/5·25-s + 1.37·26-s + 1.73·27-s + 1.67·29-s − 0.547·30-s − 0.718·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 302330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(302330\)    =    \(2 \cdot 5 \cdot 7^{2} \cdot 617\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{302330} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 302330,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $11.78776788$
$L(\frac12)$  $\approx$  $11.78776788$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7,\;617\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7,\;617\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
617 \( 1 - T \)
good3 \( 1 - p T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−12.98703490460288, −12.43691189214771, −11.76263097778840, −11.37039921116267, −10.87090496770150, −10.27558507702841, −10.14650701985108, −9.180071717013379, −8.822450548914188, −8.573095623862880, −8.164297507766800, −7.571601705994856, −7.147778064000476, −6.710877801473166, −6.119375358563173, −5.582671057386894, −4.854666597598288, −4.311664501131702, −4.013703657330667, −3.322202189446680, −3.175354774824498, −2.453861385327850, −2.068370395852065, −1.276038098681667, −0.7392818999794672, 0.7392818999794672, 1.276038098681667, 2.068370395852065, 2.453861385327850, 3.175354774824498, 3.322202189446680, 4.013703657330667, 4.311664501131702, 4.854666597598288, 5.582671057386894, 6.119375358563173, 6.710877801473166, 7.147778064000476, 7.571601705994856, 8.164297507766800, 8.573095623862880, 8.822450548914188, 9.180071717013379, 10.14650701985108, 10.27558507702841, 10.87090496770150, 11.37039921116267, 11.76263097778840, 12.43691189214771, 12.98703490460288

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