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Properties

Label 1-812-812.243-r1-0-0
Degree $1$
Conductor $812$
Sign $-0.214 + 0.976i$
Analytic cond. $87.2615$
Root an. cond. $87.2615$
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  + (0.294 − 0.955i)3-s + (0.365 − 0.930i)5-s + (−0.826 − 0.563i)9-s + (0.563 + 0.826i)11-s + (−0.900 − 0.433i)13-s + (−0.781 − 0.623i)15-s + (−0.866 − 0.5i)17-s + (−0.294 − 0.955i)19-s + (0.988 + 0.149i)23-s + (−0.733 − 0.680i)25-s + (−0.781 + 0.623i)27-s + (0.149 + 0.988i)31-s + (0.955 − 0.294i)33-s + (−0.563 + 0.826i)37-s + (−0.680 + 0.733i)39-s + ⋯
L(s)  = 1  + (0.294 − 0.955i)3-s + (0.365 − 0.930i)5-s + (−0.826 − 0.563i)9-s + (0.563 + 0.826i)11-s + (−0.900 − 0.433i)13-s + (−0.781 − 0.623i)15-s + (−0.866 − 0.5i)17-s + (−0.294 − 0.955i)19-s + (0.988 + 0.149i)23-s + (−0.733 − 0.680i)25-s + (−0.781 + 0.623i)27-s + (0.149 + 0.988i)31-s + (0.955 − 0.294i)33-s + (−0.563 + 0.826i)37-s + (−0.680 + 0.733i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $-0.214 + 0.976i$
Analytic conductor: \(87.2615\)
Root analytic conductor: \(87.2615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 812,\ (1:\ ),\ -0.214 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2013500203 - 0.2502489205i\)
\(L(\frac12)\) \(\approx\) \(-0.2013500203 - 0.2502489205i\)
\(L(1)\) \(\approx\) \(0.8434885419 - 0.4933431742i\)
\(L(1)\) \(\approx\) \(0.8434885419 - 0.4933431742i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.294 - 0.955i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
11 \( 1 + (0.563 + 0.826i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.294 - 0.955i)T \)
23 \( 1 + (0.988 + 0.149i)T \)
31 \( 1 + (0.149 + 0.988i)T \)
37 \( 1 + (-0.563 + 0.826i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.781 + 0.623i)T \)
47 \( 1 + (-0.997 + 0.0747i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.680 - 0.733i)T \)
67 \( 1 + (0.0747 - 0.997i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.930 - 0.365i)T \)
79 \( 1 + (-0.563 + 0.826i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.930 + 0.365i)T \)
97 \( 1 + (0.974 + 0.222i)T \)
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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

−22.42150261289116433951716986287, −21.729835182140519134977456824603, −21.26897847976813111594894412165, −20.264462903691636261201523374047, −19.23949929522886831582388098777, −18.93858115005090590133640370998, −17.5544732759578352101041496086, −16.98159787636682462685423503337, −16.15132945560390418196962283623, −15.12013327136661836456762476771, −14.57490205608871363592308422044, −13.98793299078833623271614216060, −12.977966235983025325789269254806, −11.628280467793767420680636593084, −11.00539496018060137758414687282, −10.238990738999469995274517290244, −9.431687799385288336914770190450, −8.68068365772875435907132825438, −7.60340466819269968345812464390, −6.49529656782462054229292715869, −5.75960176092118596984524849992, −4.58155115788246871320582449542, −3.699977793270190239068792096159, −2.83168589902527916335503368815, −1.874269123939786087565544868003, 0.0654011970912102267379290472, 1.16745060385258429266162570767, 2.087048844481937085056758479845, 3.014832928964070791887704789928, 4.56288639310792955926530500185, 5.17712790279458094321215529846, 6.48267463915958129421807572792, 7.10071308817008724886481550173, 8.068728270474876747056677704223, 9.07693680592202444959066548266, 9.447960229322068067906931248234, 10.83040604131497864131624877806, 11.95785021237972086377497893386, 12.50746285729018526835201687308, 13.217041213984985325235593247943, 13.952322102164775488361876619718, 14.90607650649822718938203530521, 15.70166686396715700066119950847, 17.02474187942670876683266145942, 17.43206920721050289638344226107, 18.04421560401213393481837495145, 19.3062024795148015275899100818, 19.82489611418460774844597646402, 20.423393458801744613169937667713, 21.3294345606345833524948951874

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