Properties

Label 1-712-712.205-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.838 - 0.545i$
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  + (−0.977 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.838 - 0.545i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.838 - 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2744195287 - 0.9252587312i\)
\(L(\frac12)\) \(\approx\) \(0.2744195287 - 0.9252587312i\)
\(L(1)\) \(\approx\) \(0.7910721749 - 0.2519689218i\)
\(L(1)\) \(\approx\) \(0.7910721749 - 0.2519689218i\)

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

−22.59523314487437483783255660336, −22.02976787873965497688282405630, −21.19996959739233779635917132630, −20.74062061051040237720263716307, −19.06200888852964999355076886446, −18.479789260249172734072304304626, −17.98440590490159162766836116697, −17.0416784143725054307739712114, −16.27495815970161750478520878526, −15.28383310697231821408934480844, −14.76771531723921471554026212194, −13.435835499359115631463014266938, −12.82085643972573847571261851928, −11.715241143037276353959576843466, −11.10909656094886386487806583816, −10.14429760479777096134074292923, −9.60008629890204139769184453782, −8.3400698011090740833349548996, −7.263526882377527080808573326346, −6.14838914590635592943938741386, −5.608884659566550089127697676360, −4.95765876893135606396809251819, −3.31491093620299091914876963517, −2.51508972004701331059500619566, −1.12274955280912195352249659027, 0.27936719428436501010005271741, 1.356960127494709218149796738312, 2.24810628171882024068192896019, 4.1303541033978073857359200220, 4.7660591291509603885740918574, 5.620520361908428638309005809923, 6.72683161088883905848850674650, 7.27179463081335881747552862946, 8.53400378369838406167210977002, 9.67311659646014602780613917559, 10.35822890016640122005769332869, 11.109715016354358769146400055565, 12.18704087116420871555968762732, 13.021708072681027222260021480653, 13.44252361077047037432062102944, 14.60744649816715005991089387678, 15.69354902466706877350155566772, 16.7579977819492083855374696695, 17.061151188735001005704164940715, 17.72299884390004593307392542465, 18.6854962447461752115467733147, 19.619863889971441219131436142021, 20.63016657324322250420688523549, 21.30736701533366070728225587040, 21.98250502107264949926502302021

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