Properties

Label 2-712-712.411-c0-0-0
Degree $2$
Conductor $712$
Sign $0.484 + 0.874i$
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  − 2-s + (1 − i)3-s + 4-s + (−1 + i)6-s − 8-s i·9-s + (1 − i)12-s + 16-s − 2i·17-s + i·18-s + (1 + i)19-s + (−1 + i)24-s − 25-s − 32-s + 2i·34-s + ⋯
L(s)  = 1  − 2-s + (1 − i)3-s + 4-s + (−1 + i)6-s − 8-s i·9-s + (1 − i)12-s + 16-s − 2i·17-s + i·18-s + (1 + i)19-s + (−1 + i)24-s − 25-s − 32-s + 2i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.484 + 0.874i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.484 + 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8571746403\)
\(L(\frac12)\) \(\approx\) \(0.8571746403\)
\(L(1)\) 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 \( 1 + (-1 + i)T - iT^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + 2iT - T^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (1 - i)T - iT^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-1 - i)T + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−10.08260456915021947744777590939, −9.556404074664540903127780764487, −8.647360074420392612853894366415, −7.87852629410409322523015715003, −7.33805583016165011502789654692, −6.54995527021202595664794663488, −5.29587821630188251337893244408, −3.37996673528472866927748458509, −2.51088620129339015279692973251, −1.34333443917995478428378572737, 1.91280228653227803732442601224, 3.16855469631222340141853472151, 3.99253737858937415505457319982, 5.43472765484756909476290493256, 6.58965031554548100563229572498, 7.68468851151705499300570148557, 8.474442016652592972407008213452, 9.015928188130892768391432844395, 9.904505525739370924721923612100, 10.36408845790350880619590272103

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