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Properties

Label 2-712-712.235-c0-0-0
Degree $2$
Conductor $712$
Sign $0.211 + 0.977i$
Analytic cond. $0.355334$
Root an. cond. $0.596099$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.142 + 0.989i)2-s + (−0.425 − 1.45i)3-s + (−0.959 − 0.281i)4-s + (1.49 − 0.215i)6-s + (0.415 − 0.909i)8-s + (−1.08 + 0.694i)9-s + (−0.345 − 0.755i)11-s + 1.51i·12-s + (0.841 + 0.540i)16-s + (−0.0405 − 0.281i)17-s + (−0.533 − 1.16i)18-s + (−0.983 − 1.53i)19-s + (0.797 − 0.234i)22-s + (−1.49 − 0.215i)24-s + (−0.654 − 0.755i)25-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (−0.425 − 1.45i)3-s + (−0.959 − 0.281i)4-s + (1.49 − 0.215i)6-s + (0.415 − 0.909i)8-s + (−1.08 + 0.694i)9-s + (−0.345 − 0.755i)11-s + 1.51i·12-s + (0.841 + 0.540i)16-s + (−0.0405 − 0.281i)17-s + (−0.533 − 1.16i)18-s + (−0.983 − 1.53i)19-s + (0.797 − 0.234i)22-s + (−1.49 − 0.215i)24-s + (−0.654 − 0.755i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.211 + 0.977i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.211 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5764331670\)
\(L(\frac12)\) \(\approx\) \(0.5764331670\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
good3 \( 1 + (0.425 + 1.45i)T + (-0.841 + 0.540i)T^{2} \)
5 \( 1 + (0.654 + 0.755i)T^{2} \)
7 \( 1 + (-0.654 - 0.755i)T^{2} \)
11 \( 1 + (0.345 + 0.755i)T + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.841 - 0.540i)T^{2} \)
17 \( 1 + (0.0405 + 0.281i)T + (-0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.983 + 1.53i)T + (-0.415 + 0.909i)T^{2} \)
23 \( 1 + (0.415 - 0.909i)T^{2} \)
29 \( 1 + (-0.654 - 0.755i)T^{2} \)
31 \( 1 + (0.415 + 0.909i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (-1.80 + 0.822i)T + (0.654 - 0.755i)T^{2} \)
47 \( 1 + (-0.841 - 0.540i)T^{2} \)
53 \( 1 + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.557 - 1.89i)T + (-0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.142 - 0.989i)T^{2} \)
67 \( 1 + (-1.61 + 0.474i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (0.654 - 0.755i)T^{2} \)
73 \( 1 + (-1.10 - 0.708i)T + (0.415 + 0.909i)T^{2} \)
79 \( 1 + (-0.415 - 0.909i)T^{2} \)
83 \( 1 + (0.557 - 0.0801i)T + (0.959 - 0.281i)T^{2} \)
97 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
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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

−10.51594221479124153415562738655, −9.217944396626661269143731513646, −8.457533164751596377588466972251, −7.64674493513547889740460095133, −6.93816984259074663347812240708, −6.20072108463558227819756497629, −5.48162755964257861449615449548, −4.25707274907096844503807342888, −2.46217377107256270752906095520, −0.70134381610135679971868089363, 2.04279115859931417850408170325, 3.55262675489948510711511456741, 4.20545005577961408594821184555, 5.08135719906434469200772858154, 5.98448955220992161223693280491, 7.65526072893338459650040287250, 8.586028248454414946452009843258, 9.601425496816409432556667870417, 9.974776885540499399015154612598, 10.76552146284997461898183114067

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