Properties

Label 2-712-712.171-c1-0-69
Degree $2$
Conductor $712$
Sign $0.0125 + 0.999i$
Analytic cond. $5.68534$
Root an. cond. $2.38439$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.126 + 1.40i)2-s + (−0.474 − 1.85i)3-s + (−1.96 − 0.355i)4-s + (1.80 − 0.392i)5-s + (2.67 − 0.433i)6-s + (0.264 − 1.46i)7-s + (0.749 − 2.72i)8-s + (−0.593 + 0.324i)9-s + (0.324 + 2.59i)10-s + (−1.47 − 2.30i)11-s + (0.272 + 3.82i)12-s + (−2.08 − 1.23i)13-s + (2.03 + 0.558i)14-s + (−1.58 − 3.16i)15-s + (3.74 + 1.40i)16-s + (0.00811 − 0.113i)17-s + ⋯
L(s)  = 1  + (−0.0893 + 0.996i)2-s + (−0.273 − 1.07i)3-s + (−0.984 − 0.177i)4-s + (0.807 − 0.175i)5-s + (1.09 − 0.176i)6-s + (0.100 − 0.554i)7-s + (0.265 − 0.964i)8-s + (−0.197 + 0.108i)9-s + (0.102 + 0.819i)10-s + (−0.445 − 0.693i)11-s + (0.0785 + 1.10i)12-s + (−0.579 − 0.343i)13-s + (0.543 + 0.149i)14-s + (−0.409 − 0.817i)15-s + (0.936 + 0.350i)16-s + (0.00196 − 0.0275i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0125 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0125 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.0125 + 0.999i$
Analytic conductor: \(5.68534\)
Root analytic conductor: \(2.38439\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :1/2),\ 0.0125 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.720148 - 0.711165i\)
\(L(\frac12)\) \(\approx\) \(0.720148 - 0.711165i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.126 - 1.40i)T \)
89 \( 1 + (4.86 - 8.08i)T \)
good3 \( 1 + (0.474 + 1.85i)T + (-2.63 + 1.43i)T^{2} \)
5 \( 1 + (-1.80 + 0.392i)T + (4.54 - 2.07i)T^{2} \)
7 \( 1 + (-0.264 + 1.46i)T + (-6.55 - 2.44i)T^{2} \)
11 \( 1 + (1.47 + 2.30i)T + (-4.56 + 10.0i)T^{2} \)
13 \( 1 + (2.08 + 1.23i)T + (6.23 + 11.4i)T^{2} \)
17 \( 1 + (-0.00811 + 0.113i)T + (-16.8 - 2.41i)T^{2} \)
19 \( 1 + (4.56 - 5.66i)T + (-4.03 - 18.5i)T^{2} \)
23 \( 1 + (0.431 + 4.01i)T + (-22.4 + 4.88i)T^{2} \)
29 \( 1 + (-3.55 - 5.12i)T + (-10.1 + 27.1i)T^{2} \)
31 \( 1 + (-0.886 + 8.24i)T + (-30.2 - 6.58i)T^{2} \)
37 \( 1 + (8.77 + 3.63i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-7.85 + 4.66i)T + (19.6 - 35.9i)T^{2} \)
43 \( 1 + (-5.62 - 3.90i)T + (15.0 + 40.2i)T^{2} \)
47 \( 1 + (7.90 + 5.91i)T + (13.2 + 45.0i)T^{2} \)
53 \( 1 + (0.735 + 0.982i)T + (-14.9 + 50.8i)T^{2} \)
59 \( 1 + (3.64 + 0.929i)T + (51.7 + 28.2i)T^{2} \)
61 \( 1 + (-0.282 + 7.91i)T + (-60.8 - 4.35i)T^{2} \)
67 \( 1 + (-1.94 + 13.5i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (-1.97 + 9.09i)T + (-64.5 - 29.4i)T^{2} \)
73 \( 1 + (-1.26 - 4.31i)T + (-61.4 + 39.4i)T^{2} \)
79 \( 1 + (-8.09 - 4.42i)T + (42.7 + 66.4i)T^{2} \)
83 \( 1 + (0.181 - 0.362i)T + (-49.7 - 66.4i)T^{2} \)
97 \( 1 + (-11.6 - 7.51i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.13877175472442930422902124378, −9.196354705330773207746136702899, −8.101268885048946976545779965032, −7.63111681180814657396632969084, −6.55733937519127324716431823770, −6.04088286056240112119974771299, −5.14429157062321426161946354598, −3.87542517278551497695007479165, −1.99673156694454805384252011871, −0.55456851168111970824392828039, 1.93934420003049333771844408940, 2.85415114851103235370306379197, 4.32949478459904422874525040859, 4.88315299157970361079232548878, 5.79578480335665409337044533225, 7.18129007193928077271255012498, 8.529222046066871208357929431539, 9.320654066595361037350848728556, 9.946241575818860320145826893232, 10.45739225468015654757564942141

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