Properties

Label 2-15631-1.1-c1-0-2
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·5-s + 3·8-s − 3·9-s − 2·10-s − 11-s − 6·13-s − 16-s + 2·17-s + 3·18-s + 8·19-s − 2·20-s + 22-s − 25-s + 6·26-s + 29-s + 4·31-s − 5·32-s − 2·34-s + 3·36-s − 2·37-s − 8·38-s + 6·40-s + 2·41-s − 4·43-s + 44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 9-s − 0.632·10-s − 0.301·11-s − 1.66·13-s − 1/4·16-s + 0.485·17-s + 0.707·18-s + 1.83·19-s − 0.447·20-s + 0.213·22-s − 1/5·25-s + 1.17·26-s + 0.185·29-s + 0.718·31-s − 0.883·32-s − 0.342·34-s + 1/2·36-s − 0.328·37-s − 1.29·38-s + 0.948·40-s + 0.312·41-s − 0.609·43-s + 0.150·44-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: \(1\)
Selberg data: \((2,\ 15631,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−16.62731643046730, −15.76729868714415, −15.22947862052280, −14.24088655975242, −14.04052995733013, −13.85961358274411, −12.89424707950879, −12.40321271405921, −11.74271753253412, −11.16401806163391, −10.29493779480633, −9.938935269355362, −9.429806216036498, −9.127293599759417, −8.122573787063101, −7.834735273854200, −7.187331444534128, −6.342195512496106, −5.417484486881167, −5.250318978117278, −4.543992065678343, −3.392546514676651, −2.745788818152952, −1.950839446978662, −0.9778534176548871, 0, 0.9778534176548871, 1.950839446978662, 2.745788818152952, 3.392546514676651, 4.543992065678343, 5.250318978117278, 5.417484486881167, 6.342195512496106, 7.187331444534128, 7.834735273854200, 8.122573787063101, 9.127293599759417, 9.429806216036498, 9.938935269355362, 10.29493779480633, 11.16401806163391, 11.74271753253412, 12.40321271405921, 12.89424707950879, 13.85961358274411, 14.04052995733013, 14.24088655975242, 15.22947862052280, 15.76729868714415, 16.62731643046730

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