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Properties

Label 1-712-712.507-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.155 - 0.987i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + (−0.977 − 0.212i)3-s + (0.755 − 0.654i)5-s + (0.0713 − 0.997i)7-s + (0.909 + 0.415i)9-s + (0.654 − 0.755i)11-s + (0.212 − 0.977i)13-s + (−0.877 + 0.479i)15-s + (0.281 + 0.959i)17-s + (−0.349 − 0.936i)19-s + (−0.281 + 0.959i)21-s + (0.936 − 0.349i)23-s + (0.142 − 0.989i)25-s + (−0.800 − 0.599i)27-s + (0.997 + 0.0713i)29-s + (0.936 + 0.349i)31-s + ⋯
L(s)  = 1  + (−0.977 − 0.212i)3-s + (0.755 − 0.654i)5-s + (0.0713 − 0.997i)7-s + (0.909 + 0.415i)9-s + (0.654 − 0.755i)11-s + (0.212 − 0.977i)13-s + (−0.877 + 0.479i)15-s + (0.281 + 0.959i)17-s + (−0.349 − 0.936i)19-s + (−0.281 + 0.959i)21-s + (0.936 − 0.349i)23-s + (0.142 − 0.989i)25-s + (−0.800 − 0.599i)27-s + (0.997 + 0.0713i)29-s + (0.936 + 0.349i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.155 - 0.987i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.155 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8373615978 - 0.9792273843i\)
\(L(\frac12)\) \(\approx\) \(0.8373615978 - 0.9792273843i\)
\(L(1)\) \(\approx\) \(0.9148690117 - 0.4092658740i\)
\(L(1)\) \(\approx\) \(0.9148690117 - 0.4092658740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.977 - 0.212i)T \)
5 \( 1 + (0.755 - 0.654i)T \)
7 \( 1 + (0.0713 - 0.997i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (0.212 - 0.977i)T \)
17 \( 1 + (0.281 + 0.959i)T \)
19 \( 1 + (-0.349 - 0.936i)T \)
23 \( 1 + (0.936 - 0.349i)T \)
29 \( 1 + (0.997 + 0.0713i)T \)
31 \( 1 + (0.936 + 0.349i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (0.212 + 0.977i)T \)
43 \( 1 + (-0.997 + 0.0713i)T \)
47 \( 1 + (-0.540 + 0.841i)T \)
53 \( 1 + (0.540 + 0.841i)T \)
59 \( 1 + (0.977 - 0.212i)T \)
61 \( 1 + (-0.599 + 0.800i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.755 - 0.654i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (0.909 - 0.415i)T \)
83 \( 1 + (-0.877 - 0.479i)T \)
97 \( 1 + (-0.654 - 0.755i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.832307674474608640117274563240, −22.03927574237604517066802355634, −21.27781487442402007965820202954, −20.86303547281760300914014606362, −19.21466101785951686665049372480, −18.64344791598494571885471267914, −17.88001832177955722262031034215, −17.22836040282685837149577584609, −16.37496025602291380280510750017, −15.45025580099183155128672192563, −14.66841618094729687886903155049, −13.84051653978066258415250148891, −12.66251299798351295669575999141, −11.85257610445035849258889631218, −11.33740243955054851350734957776, −10.130287542937693138962866366815, −9.62595568786259682456214779813, −8.689017519401941848442091826304, −7.05332507595636292247217673563, −6.59303471920326127458141296077, −5.63578737251559719337160599387, −4.89996045903899231524942286435, −3.71982495629253767388489571745, −2.36200771217833755911979850136, −1.44696878892067827652551514951, 0.82485526816583071383276911286, 1.3909472747348820269423778397, 3.07751676159720435466227911553, 4.41440117365014899722125111197, 5.08338263889584805814421932565, 6.198762071967940994211241469825, 6.65602902771274214676669490524, 7.967163071189097824355424068162, 8.81780826761232345560426071001, 10.117544482660842041138955212494, 10.56025942856344276965900201702, 11.48579218772823854282477203028, 12.5322073171035350535077405881, 13.23340717107989482492197862302, 13.79988453637934649849813845539, 15.03150572376710967786469420791, 16.15298334343062474315213483113, 16.87816713238068932211777489308, 17.359052887731559976338909946354, 17.95256556500541573732422217241, 19.203920958213656959253347146124, 19.86682055088619318688729427850, 20.97895725619151870528867309680, 21.5466362082694248802907611442, 22.40799637460143973406559333957

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