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Properties

Label 1-712-712.293-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.953 - 0.302i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
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  + (0.936 − 0.349i)3-s + (−0.989 − 0.142i)5-s + (0.800 − 0.599i)7-s + (0.755 − 0.654i)9-s + (−0.142 − 0.989i)11-s + (−0.349 − 0.936i)13-s + (−0.977 + 0.212i)15-s + (−0.540 − 0.841i)17-s + (0.997 + 0.0713i)19-s + (0.540 − 0.841i)21-s + (−0.0713 + 0.997i)23-s + (0.959 + 0.281i)25-s + (0.479 − 0.877i)27-s + (−0.599 − 0.800i)29-s + (−0.0713 − 0.997i)31-s + ⋯
L(s)  = 1  + (0.936 − 0.349i)3-s + (−0.989 − 0.142i)5-s + (0.800 − 0.599i)7-s + (0.755 − 0.654i)9-s + (−0.142 − 0.989i)11-s + (−0.349 − 0.936i)13-s + (−0.977 + 0.212i)15-s + (−0.540 − 0.841i)17-s + (0.997 + 0.0713i)19-s + (0.540 − 0.841i)21-s + (−0.0713 + 0.997i)23-s + (0.959 + 0.281i)25-s + (0.479 − 0.877i)27-s + (−0.599 − 0.800i)29-s + (−0.0713 − 0.997i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.953 - 0.302i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.953 - 0.302i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2967119836 - 1.918144719i\)
\(L(\frac12)\) \(\approx\) \(0.2967119836 - 1.918144719i\)
\(L(1)\) \(\approx\) \(1.116534872 - 0.5791046327i\)
\(L(1)\) \(\approx\) \(1.116534872 - 0.5791046327i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.936 - 0.349i)T \)
5 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (0.800 - 0.599i)T \)
11 \( 1 + (-0.142 - 0.989i)T \)
13 \( 1 + (-0.349 - 0.936i)T \)
17 \( 1 + (-0.540 - 0.841i)T \)
19 \( 1 + (0.997 + 0.0713i)T \)
23 \( 1 + (-0.0713 + 0.997i)T \)
29 \( 1 + (-0.599 - 0.800i)T \)
31 \( 1 + (-0.0713 - 0.997i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.349 + 0.936i)T \)
43 \( 1 + (-0.599 + 0.800i)T \)
47 \( 1 + (0.909 - 0.415i)T \)
53 \( 1 + (0.909 + 0.415i)T \)
59 \( 1 + (-0.936 - 0.349i)T \)
61 \( 1 + (0.877 + 0.479i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
71 \( 1 + (-0.989 + 0.142i)T \)
73 \( 1 + (0.654 - 0.755i)T \)
79 \( 1 + (-0.755 - 0.654i)T \)
83 \( 1 + (-0.977 - 0.212i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−22.58541696449747536107015280992, −21.910529886024199862245234230469, −21.018274291348369797656010695725, −20.25452991268303338609622415582, −19.679974697379596775899317424690, −18.74910614884173283502337842889, −18.15576472268728868316232731517, −16.94967414386317080242019808863, −15.87244960471368213380259659762, −15.340099808129331709741581614499, −14.5861452561911014195702226529, −14.04852140741426728207245902678, −12.64022376041154178827122594304, −12.08119073737031311997408761630, −11.01865864281036540804815835173, −10.21081151181167613909419442903, −8.96029638287416272682333373099, −8.584273220487219580466491196815, −7.47137330279488863836451507191, −6.96521691378539549886448599371, −5.18335131613881401650472914753, −4.450658251566777770776843659766, −3.645734182101226571197726271894, −2.4349119920347467612423519218, −1.63493664026525373080006472462, 0.39384029657601761885522655968, 1.3045061205391013456191300155, 2.77659753858713873611967968081, 3.5310911618715093682159023168, 4.46888321456235305901159423013, 5.544713140605827617394370636990, 7.06725061855997027784045080415, 7.73315349015687578872414345148, 8.19705396399133286980943294621, 9.198043638500317218019976885536, 10.242871519412154350788618193723, 11.4157731739060091073424255061, 11.85107065386804890220571132855, 13.27765467458170774446774763585, 13.588139374338965285301479130942, 14.67853059295777153877961647344, 15.34927823614924488265045092970, 16.095402785868386350722611749896, 17.178086445216572512329831194767, 18.18262200594228598035167142640, 18.828739074782361875598491075755, 19.80203890931936429519382586114, 20.2643620901657837238686731103, 20.89270147248056379162794771770, 22.00262813895635223140903581687

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