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Properties

Label 1-812-812.499-r1-0-0
Degree $1$
Conductor $812$
Sign $0.954 - 0.297i$
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  + (0.826 − 0.563i)3-s + (−0.733 + 0.680i)5-s + (0.365 − 0.930i)9-s + (0.365 + 0.930i)11-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)15-s + (0.5 − 0.866i)17-s + (0.826 + 0.563i)19-s + (−0.955 + 0.294i)23-s + (0.0747 − 0.997i)25-s + (−0.222 − 0.974i)27-s + (0.955 + 0.294i)31-s + (0.826 + 0.563i)33-s + (−0.365 + 0.930i)37-s + (0.0747 − 0.997i)39-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)3-s + (−0.733 + 0.680i)5-s + (0.365 − 0.930i)9-s + (0.365 + 0.930i)11-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)15-s + (0.5 − 0.866i)17-s + (0.826 + 0.563i)19-s + (−0.955 + 0.294i)23-s + (0.0747 − 0.997i)25-s + (−0.222 − 0.974i)27-s + (0.955 + 0.294i)31-s + (0.826 + 0.563i)33-s + (−0.365 + 0.930i)37-s + (0.0747 − 0.997i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $0.954 - 0.297i$
Analytic conductor: \(87.2615\)
Root analytic conductor: \(87.2615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 812,\ (1:\ ),\ 0.954 - 0.297i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.726547545 - 0.4147982639i\)
\(L(\frac12)\) \(\approx\) \(2.726547545 - 0.4147982639i\)
\(L(1)\) \(\approx\) \(1.389366911 - 0.1297510615i\)
\(L(1)\) \(\approx\) \(1.389366911 - 0.1297510615i\)

Euler product

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

−21.86587984861963865332594839827, −21.156575861197836374925507628810, −20.51892257140349797036893994675, −19.54450610472888516514435326224, −19.27118621858261309689931265739, −18.24580710644913079974657333189, −16.86088385774775593734465769311, −16.32804004470722078326176694749, −15.685901770809493621688839787452, −14.84448156046016310095889233299, −13.874730198332575062470189005921, −13.37857670449376956727966472873, −12.149807804489142525100743785188, −11.439591733900581070532484162711, −10.47110899193200014264522695355, −9.470037473400824048255609494390, −8.60357100479400321634514511025, −8.24486900573394730958789489836, −7.15582523325989211672234061788, −5.90025392504481826444585671499, −4.83312926533137042298968957708, −3.866480477588496926452703722074, −3.42045676557138154394556492886, −1.98096251130142595595409509013, −0.812529138021652058626242530540, 0.775231516543160490828359536146, 1.91715896736592926078089560612, 3.12417312305440307980595321507, 3.59401536914996353251529231094, 4.82776114654238516346829317172, 6.23492370986691074818535571075, 7.03452034317596677388981649101, 7.823617134747396485410335593363, 8.35949103306071637429090908039, 9.6838495962995398098586087620, 10.20169303164633159532856176039, 11.66813891786236382102360711692, 12.0279718291459754206675207694, 13.092155111346840538806568905341, 14.00666304774486857926457641071, 14.599326730564468220755704779894, 15.47035662020913656481763417988, 16.00977829572899664576199788330, 17.44155179919502326308911130538, 18.28691948926630720000477867207, 18.648478230660666733814832027868, 19.74489405778603086239741842226, 20.19940586953527790841148971153, 20.89579007652447330922431130910, 22.14716998000314920373055170496

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