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Properties

Label 1-712-712.83-r0-0-0
Degree $1$
Conductor $712$
Sign $0.980 + 0.196i$
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  + (−0.349 − 0.936i)3-s + (0.989 + 0.142i)5-s + (−0.599 − 0.800i)7-s + (−0.755 + 0.654i)9-s + (0.142 + 0.989i)11-s + (−0.936 + 0.349i)13-s + (−0.212 − 0.977i)15-s + (0.540 + 0.841i)17-s + (−0.0713 + 0.997i)19-s + (−0.540 + 0.841i)21-s + (0.997 + 0.0713i)23-s + (0.959 + 0.281i)25-s + (0.877 + 0.479i)27-s + (−0.800 + 0.599i)29-s + (0.997 − 0.0713i)31-s + ⋯
L(s)  = 1  + (−0.349 − 0.936i)3-s + (0.989 + 0.142i)5-s + (−0.599 − 0.800i)7-s + (−0.755 + 0.654i)9-s + (0.142 + 0.989i)11-s + (−0.936 + 0.349i)13-s + (−0.212 − 0.977i)15-s + (0.540 + 0.841i)17-s + (−0.0713 + 0.997i)19-s + (−0.540 + 0.841i)21-s + (0.997 + 0.0713i)23-s + (0.959 + 0.281i)25-s + (0.877 + 0.479i)27-s + (−0.800 + 0.599i)29-s + (0.997 − 0.0713i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.980 + 0.196i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.980 + 0.196i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.240296819 + 0.1231876014i\)
\(L(\frac12)\) \(\approx\) \(1.240296819 + 0.1231876014i\)
\(L(1)\) \(\approx\) \(1.011427138 - 0.1153237808i\)
\(L(1)\) \(\approx\) \(1.011427138 - 0.1153237808i\)

Euler product

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

−22.248297759005968381489102563451, −21.85488393797914544740834766903, −21.16454588574868541599894291123, −20.37214416698981086238055172816, −19.28435933563561062697502091745, −18.49418461491071947870637023638, −17.39663453169800524818803342350, −16.92160630946488535199184982317, −16.04245290699283900395791615602, −15.2909342360514412559416983836, −14.42520011047474976031143475249, −13.52798804879588603136301267368, −12.58821981264256863846201271971, −11.65698670264940438773405307928, −10.80374726962169020474245344632, −9.77623674062238599020706787929, −9.33223668562524005375824386189, −8.576871638557181628273406365383, −7.02234498889931594478656213019, −5.93676327631518590503876452705, −5.45818114907251583642689449240, −4.56716340826452569647066230430, −3.05537866022711082316375250311, −2.60008358461416954683584411564, −0.67923064045381622941823608812, 1.22716356570934096870363018233, 2.03389734528785115481801862204, 3.1517683571216735182154743498, 4.56682595250926537238832171258, 5.597259611175172640383690977617, 6.51878073035651710082993017626, 7.09883511160504966338050502701, 7.96913122893708516860747843500, 9.3428453684035197333215278344, 10.07208990453146728139569792195, 10.80584610752614648761143726961, 12.106403756325022927087854043653, 12.71409539202960091200425653032, 13.40166336532612585361853503134, 14.290931309775843631788189695238, 14.92728257000314439194851183958, 16.55500469717421341841247022192, 17.080756367458324580550459765714, 17.51924816616104387065134234440, 18.64456913346866015714458711748, 19.20115669788014890785884044120, 20.13975368527952691819420006857, 20.924787571969691194206297761664, 22.06060832742286564259798172325, 22.66343768722788777395592438674

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