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Properties

Label 1-837-837.2-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.0729 - 0.997i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (−0.0348 + 0.999i)2-s + (−0.997 − 0.0697i)4-s + (−0.173 + 0.984i)5-s + (−0.241 + 0.970i)7-s + (0.104 − 0.994i)8-s + (−0.978 − 0.207i)10-s + (0.719 + 0.694i)11-s + (0.0348 + 0.999i)13-s + (−0.961 − 0.275i)14-s + (0.990 + 0.139i)16-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (0.241 − 0.970i)20-s + (−0.719 + 0.694i)22-s + (0.241 + 0.970i)23-s + ⋯
L(s)  = 1  + (−0.0348 + 0.999i)2-s + (−0.997 − 0.0697i)4-s + (−0.173 + 0.984i)5-s + (−0.241 + 0.970i)7-s + (0.104 − 0.994i)8-s + (−0.978 − 0.207i)10-s + (0.719 + 0.694i)11-s + (0.0348 + 0.999i)13-s + (−0.961 − 0.275i)14-s + (0.990 + 0.139i)16-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (0.241 − 0.970i)20-s + (−0.719 + 0.694i)22-s + (0.241 + 0.970i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0729 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0729 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.0729 - 0.997i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.0729 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9189706203 + 0.9886255006i\)
\(L(\frac12)\) \(\approx\) \(-0.9189706203 + 0.9886255006i\)
\(L(1)\) \(\approx\) \(0.4445759360 + 0.7797647852i\)
\(L(1)\) \(\approx\) \(0.4445759360 + 0.7797647852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.0348 + 0.999i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.241 + 0.970i)T \)
11 \( 1 + (0.719 + 0.694i)T \)
13 \( 1 + (0.0348 + 0.999i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.241 + 0.970i)T \)
29 \( 1 + (-0.0348 + 0.999i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.615 + 0.788i)T \)
43 \( 1 + (-0.882 + 0.469i)T \)
47 \( 1 + (0.374 + 0.927i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.882 + 0.469i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (0.559 + 0.829i)T \)
83 \( 1 + (-0.0348 + 0.999i)T \)
89 \( 1 + (-0.913 - 0.406i)T \)
97 \( 1 + (-0.719 - 0.694i)T \)
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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

−20.94601685161668210044699577848, −20.60469742153327036897660602982, −19.79907697741668942663446560373, −19.35472027532275212414954225686, −18.20538581690439125921904759221, −17.37374912010673443349935118491, −16.72086209310249473881741027769, −15.93584833043563385172330562196, −14.644633392808864164160195628047, −13.64226971371328047581457742569, −13.254127459100742614232769673421, −12.30620284456472305696425795079, −11.59777590133017901429656654033, −10.66078079488508745847085477462, −9.93677359406790906490526484262, −8.96955349920763424556574173828, −8.34027277326954219305057776826, −7.33693046944177872694565866977, −5.93428595384866259106880426807, −4.976951214839201797872241491227, −4.0265708231955513273822385418, −3.4409430308976043104049137338, −2.076517587833207726145528697052, −0.797803949870367822763647121124, −0.424614245049436717921602818170, 1.54755697327991206181452029067, 2.843917341934960043611369176850, 3.916220354880389340246484682055, 4.86328476501979805517191783631, 5.953055747769036497521504662427, 6.78496450868321706277835188120, 7.17719444085998015543454320448, 8.456953551719230381768443025356, 9.254923302258968316540936434070, 9.84463395196795193342053541730, 11.18885992744832309134179644511, 11.91187056325402678551481291209, 12.98279521606786980668103781685, 13.89594546784701585965085770720, 14.673300913701315841694611142583, 15.30398173223603661114486992406, 15.870985182953723937429965827180, 16.91249302304509790675978812384, 17.8610920170162712793879468719, 18.26185264315692138617563598147, 19.26069108245351858272786963115, 19.72039970212833237876165580736, 21.34789178128820776139446024260, 22.16993475936965454880922189986, 22.32368126033225624157559393292

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