Properties

Label 2-12936-1.1-c1-0-4
Degree $2$
Conductor $12936$
Sign $1$
Analytic cond. $103.294$
Root an. cond. $10.1633$
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  + 3-s − 2·5-s + 9-s + 11-s − 2·13-s − 2·15-s − 6·17-s + 4·23-s − 25-s + 27-s + 2·29-s + 33-s − 10·37-s − 2·39-s − 6·41-s − 8·43-s − 2·45-s + 4·47-s − 6·51-s − 6·53-s − 2·55-s + 12·59-s − 2·61-s + 4·65-s + 4·67-s + 4·69-s + 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.174·33-s − 1.64·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.298·45-s + 0.583·47-s − 0.840·51-s − 0.824·53-s − 0.269·55-s + 1.56·59-s − 0.256·61-s + 0.496·65-s + 0.488·67-s + 0.481·69-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12936\)    =    \(2^{3} \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(103.294\)
Root analytic conductor: \(10.1633\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 12936,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.548290530\)
\(L(\frac12)\) \(\approx\) \(1.548290530\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.96418956693212, −15.71160386681829, −15.07073396088244, −14.83231532656858, −13.83025801700200, −13.66869024907895, −12.82737948849049, −12.30225305837159, −11.73723944966493, −11.15500920249618, −10.59152789981063, −9.841516372949575, −9.230371083402422, −8.597038650238980, −8.202220359699161, −7.451601597868428, −6.809652690086707, −6.476122037438251, −5.119374981305609, −4.834152871869076, −3.826967474850850, −3.508855986237385, −2.490585048244883, −1.808614525456164, −0.5254636043949540, 0.5254636043949540, 1.808614525456164, 2.490585048244883, 3.508855986237385, 3.826967474850850, 4.834152871869076, 5.119374981305609, 6.476122037438251, 6.809652690086707, 7.451601597868428, 8.202220359699161, 8.597038650238980, 9.230371083402422, 9.841516372949575, 10.59152789981063, 11.15500920249618, 11.73723944966493, 12.30225305837159, 12.82737948849049, 13.66869024907895, 13.83025801700200, 14.83231532656858, 15.07073396088244, 15.71160386681829, 15.96418956693212

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