Properties

Degree 2
Conductor $ 2 \cdot 13 \cdot 1171 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s − 7-s − 8-s + 9-s + 3·11-s − 2·12-s + 13-s + 14-s + 16-s − 3·17-s − 18-s + 2·19-s + 2·21-s − 3·22-s − 3·23-s + 2·24-s − 5·25-s − 26-s + 4·27-s − 28-s − 4·31-s − 32-s − 6·33-s + 3·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.577·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.458·19-s + 0.436·21-s − 0.639·22-s − 0.625·23-s + 0.408·24-s − 25-s − 0.196·26-s + 0.769·27-s − 0.188·28-s − 0.718·31-s − 0.176·32-s − 1.04·33-s + 0.514·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30446\)    =    \(2 \cdot 13 \cdot 1171\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{30446} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 30446,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;13,\;1171\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;13,\;1171\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
13 \( 1 - T \)
1171 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−15.79128749568019, −14.92386397925478, −14.34139632278437, −13.89092390298417, −13.03636561267707, −12.63162839590496, −11.98754782603600, −11.48063124282411, −11.22062596121268, −10.66578338583999, −9.945204007910299, −9.535059761517382, −8.985476420329441, −8.367768176773600, −7.663799986440690, −7.068985478271498, −6.442938370324703, −6.040039806048648, −5.599169078416441, −4.723377322167726, −4.048709703692319, −3.377601852788726, −2.432979090708622, −1.632003573610394, −0.7959867883163105, 0, 0.7959867883163105, 1.632003573610394, 2.432979090708622, 3.377601852788726, 4.048709703692319, 4.723377322167726, 5.599169078416441, 6.040039806048648, 6.442938370324703, 7.068985478271498, 7.663799986440690, 8.367768176773600, 8.985476420329441, 9.535059761517382, 9.945204007910299, 10.66578338583999, 11.22062596121268, 11.48063124282411, 11.98754782603600, 12.63162839590496, 13.03636561267707, 13.89092390298417, 14.34139632278437, 14.92386397925478, 15.79128749568019

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