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Properties

Label 2-712-712.587-c0-0-0
Degree $2$
Conductor $712$
Sign $0.900 + 0.434i$
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  + (−0.841 − 0.540i)2-s + (−0.898 + 0.334i)3-s + (0.415 + 0.909i)4-s + (0.936 + 0.203i)6-s + (0.142 − 0.989i)8-s + (−0.0614 + 0.0532i)9-s + (−0.281 − 1.95i)11-s + (−0.677 − 0.677i)12-s + (−0.654 + 0.755i)16-s + (0.909 + 1.41i)17-s + (0.0804 − 0.0115i)18-s + (1.59 + 0.114i)19-s + (−0.822 + 1.80i)22-s + (0.203 + 0.936i)24-s + (0.959 + 0.281i)25-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)2-s + (−0.898 + 0.334i)3-s + (0.415 + 0.909i)4-s + (0.936 + 0.203i)6-s + (0.142 − 0.989i)8-s + (−0.0614 + 0.0532i)9-s + (−0.281 − 1.95i)11-s + (−0.677 − 0.677i)12-s + (−0.654 + 0.755i)16-s + (0.909 + 1.41i)17-s + (0.0804 − 0.0115i)18-s + (1.59 + 0.114i)19-s + (−0.822 + 1.80i)22-s + (0.203 + 0.936i)24-s + (0.959 + 0.281i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.900 + 0.434i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.900 + 0.434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4839067485\)
\(L(\frac12)\) \(\approx\) \(0.4839067485\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 + 0.540i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
good3 \( 1 + (0.898 - 0.334i)T + (0.755 - 0.654i)T^{2} \)
5 \( 1 + (-0.959 - 0.281i)T^{2} \)
7 \( 1 + (-0.281 + 0.959i)T^{2} \)
11 \( 1 + (0.281 + 1.95i)T + (-0.959 + 0.281i)T^{2} \)
13 \( 1 + (0.755 - 0.654i)T^{2} \)
17 \( 1 + (-0.909 - 1.41i)T + (-0.415 + 0.909i)T^{2} \)
19 \( 1 + (-1.59 - 0.114i)T + (0.989 + 0.142i)T^{2} \)
23 \( 1 + (0.989 + 0.142i)T^{2} \)
29 \( 1 + (0.281 - 0.959i)T^{2} \)
31 \( 1 + (0.989 - 0.142i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-0.494 + 1.32i)T + (-0.755 - 0.654i)T^{2} \)
43 \( 1 + (0.254 - 0.340i)T + (-0.281 - 0.959i)T^{2} \)
47 \( 1 + (-0.654 + 0.755i)T^{2} \)
53 \( 1 + (-0.654 - 0.755i)T^{2} \)
59 \( 1 + (-1.83 - 0.682i)T + (0.755 + 0.654i)T^{2} \)
61 \( 1 + (-0.540 - 0.841i)T^{2} \)
67 \( 1 + (0.627 - 1.37i)T + (-0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.959 + 0.281i)T^{2} \)
73 \( 1 + (0.368 - 0.425i)T + (-0.142 - 0.989i)T^{2} \)
79 \( 1 + (-0.142 - 0.989i)T^{2} \)
83 \( 1 + (1.83 + 0.398i)T + (0.909 + 0.415i)T^{2} \)
97 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
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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

−10.54003315492034300870955027453, −10.06235927106212159631074683210, −8.774911564090501917173716653400, −8.281738738744971941672576164281, −7.23417666011198440664840914594, −5.96027781307994632695269962782, −5.42553091724254069422172254012, −3.80060257847199500660281641743, −2.92525787369017564359103344239, −1.02009823127570854373240134823, 1.15124353710273176668984306261, 2.75509493338287393520912883840, 4.88431104164667871794305847055, 5.33535517056140593490977889997, 6.53420047338117966462181135731, 7.23991366740135584757093530257, 7.77587488089177383434662287436, 9.209817318378204162807319747187, 9.725017415606647936921783529321, 10.51570410812578686644473533741

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