Properties

Label 2-712-712.139-c0-0-0
Degree $2$
Conductor $712$
Sign $-0.965 + 0.259i$
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  + (0.415 + 0.909i)2-s + (−0.817 + 0.708i)3-s + (−0.654 + 0.755i)4-s + (−0.983 − 0.449i)6-s + (−0.959 − 0.281i)8-s + (0.0240 − 0.167i)9-s + (−1.84 + 0.540i)11-s − 1.08i·12-s + (−0.142 − 0.989i)16-s + (−0.345 + 0.755i)17-s + (0.162 − 0.0476i)18-s + (0.557 + 0.0801i)19-s + (−1.25 − 1.45i)22-s + (0.983 − 0.449i)24-s + (0.841 + 0.540i)25-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (−0.817 + 0.708i)3-s + (−0.654 + 0.755i)4-s + (−0.983 − 0.449i)6-s + (−0.959 − 0.281i)8-s + (0.0240 − 0.167i)9-s + (−1.84 + 0.540i)11-s − 1.08i·12-s + (−0.142 − 0.989i)16-s + (−0.345 + 0.755i)17-s + (0.162 − 0.0476i)18-s + (0.557 + 0.0801i)19-s + (−1.25 − 1.45i)22-s + (0.983 − 0.449i)24-s + (0.841 + 0.540i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.965 + 0.259i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ -0.965 + 0.259i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5932223928\)
\(L(\frac12)\) \(\approx\) \(0.5932223928\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
good3 \( 1 + (0.817 - 0.708i)T + (0.142 - 0.989i)T^{2} \)
5 \( 1 + (-0.841 - 0.540i)T^{2} \)
7 \( 1 + (0.841 + 0.540i)T^{2} \)
11 \( 1 + (1.84 - 0.540i)T + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (-0.142 + 0.989i)T^{2} \)
17 \( 1 + (0.345 - 0.755i)T + (-0.654 - 0.755i)T^{2} \)
19 \( 1 + (-0.557 - 0.0801i)T + (0.959 + 0.281i)T^{2} \)
23 \( 1 + (-0.959 - 0.281i)T^{2} \)
29 \( 1 + (0.841 + 0.540i)T^{2} \)
31 \( 1 + (-0.959 + 0.281i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (-0.512 - 1.74i)T + (-0.841 + 0.540i)T^{2} \)
47 \( 1 + (0.142 + 0.989i)T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (-1.37 - 1.19i)T + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.415 - 0.909i)T^{2} \)
67 \( 1 + (0.186 + 0.215i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (-0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \)
79 \( 1 + (0.959 - 0.281i)T^{2} \)
83 \( 1 + (-1.37 - 0.627i)T + (0.654 + 0.755i)T^{2} \)
97 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
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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

−10.97549261617432153993371811459, −10.28071223832763115933084990467, −9.446074469758811757568014160911, −8.245146105878434160079152836599, −7.61837504301944182300903618559, −6.54971204050487671242669673072, −5.47524655459449591453044983089, −5.06429944505356698826997645766, −4.13466401347863861677729704703, −2.75052503911364992345911355660, 0.60594101411000472404777510877, 2.30832069941141075554432547656, 3.33775286241059952292018196704, 4.90412667634016539490134941388, 5.45419394893330481174226027663, 6.40794513410563363486757220788, 7.46461148737927963998424235730, 8.536297035783742887918340113425, 9.548862043208847634458000720397, 10.57939059753442529181276709437

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