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Properties

Label 2-15631-1.1-c1-0-0
Degree $2$
Conductor $15631$
Sign $1$
Analytic cond. $124.814$
Root an. cond. $11.1720$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 2·4-s − 5-s − 4·6-s + 9-s − 2·10-s + 11-s − 4·12-s + 13-s + 2·15-s − 4·16-s + 4·17-s + 2·18-s + 8·19-s − 2·20-s + 2·22-s − 4·25-s + 2·26-s + 4·27-s + 29-s + 4·30-s − 6·31-s − 8·32-s − 2·33-s + 8·34-s + 2·36-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s − 0.447·5-s − 1.63·6-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 1.15·12-s + 0.277·13-s + 0.516·15-s − 16-s + 0.970·17-s + 0.471·18-s + 1.83·19-s − 0.447·20-s + 0.426·22-s − 4/5·25-s + 0.392·26-s + 0.769·27-s + 0.185·29-s + 0.730·30-s − 1.07·31-s − 1.41·32-s − 0.348·33-s + 1.37·34-s + 1/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15631 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15631 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15631\)    =    \(7^{2} \cdot 11 \cdot 29\)
Sign: $1$
Analytic conductor: \(124.814\)
Root analytic conductor: \(11.1720\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15631,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.302007765\)
\(L(\frac12)\) \(\approx\) \(2.302007765\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 - T \)
29 \( 1 - T \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.07279315979421, −15.41608488224734, −14.82067984460653, −14.19949182851825, −13.82931623366381, −13.22766244578696, −12.43136638597335, −12.15431874635451, −11.62759084787773, −11.34334162415725, −10.65805778273767, −9.849504682054511, −9.324648625249699, −8.391966609645684, −7.718437929758954, −6.890768361453121, −6.548210458946187, −5.635746090529559, −5.312173480992862, −5.004739455038234, −3.803237633450814, −3.661681452123049, −2.805836990438102, −1.611407400504493, −0.5683894836751541, 0.5683894836751541, 1.611407400504493, 2.805836990438102, 3.661681452123049, 3.803237633450814, 5.004739455038234, 5.312173480992862, 5.635746090529559, 6.548210458946187, 6.890768361453121, 7.718437929758954, 8.391966609645684, 9.324648625249699, 9.849504682054511, 10.65805778273767, 11.34334162415725, 11.62759084787773, 12.15431874635451, 12.43136638597335, 13.22766244578696, 13.82931623366381, 14.19949182851825, 14.82067984460653, 15.41608488224734, 16.07279315979421

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