Properties

Label 2-712-712.355-c0-0-2
Degree $2$
Conductor $712$
Sign $1$
Analytic cond. $0.355334$
Root an. cond. $0.596099$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 9-s − 2·11-s + 16-s − 2·17-s + 18-s − 2·22-s + 25-s + 32-s − 2·34-s + 36-s − 2·44-s − 49-s + 50-s + 64-s − 2·67-s − 2·68-s + 72-s − 2·73-s + 81-s − 2·88-s + 89-s + 2·97-s − 98-s − 2·99-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s + 9-s − 2·11-s + 16-s − 2·17-s + 18-s − 2·22-s + 25-s + 32-s − 2·34-s + 36-s − 2·44-s − 49-s + 50-s + 64-s − 2·67-s − 2·68-s + 72-s − 2·73-s + 81-s − 2·88-s + 89-s + 2·97-s − 98-s − 2·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $1$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{712} (355, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.659325179\)
\(L(\frac12)\) \(\approx\) \(1.659325179\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
89 \( 1 - T \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
11 \( ( 1 + T )^{2} \)
13 \( 1 + T^{2} \)
17 \( ( 1 + T )^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( ( 1 + T )^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 + T )^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )^{2} \)
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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

−10.64761043687724251625906427658, −10.21386577237434243106758962493, −8.839861704408460965258447049669, −7.73830901771754932425045716756, −7.05704020258261041489302509399, −6.12742109873435187545521876845, −4.94035479646313060351758688740, −4.46336663586957730740103762313, −3.03369951377243322332441281224, −2.04047517617842890678534652430, 2.04047517617842890678534652430, 3.03369951377243322332441281224, 4.46336663586957730740103762313, 4.94035479646313060351758688740, 6.12742109873435187545521876845, 7.05704020258261041489302509399, 7.73830901771754932425045716756, 8.839861704408460965258447049669, 10.21386577237434243106758962493, 10.64761043687724251625906427658

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