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Properties

Label 1-1157-1157.318-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.200 - 0.979i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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  + (−0.371 − 0.928i)2-s + (−0.0237 − 0.999i)3-s + (−0.723 + 0.690i)4-s + (−0.415 + 0.909i)5-s + (−0.919 + 0.393i)6-s + (0.986 + 0.165i)7-s + (0.909 + 0.415i)8-s + (−0.998 + 0.0475i)9-s + (0.998 + 0.0475i)10-s + (−0.814 − 0.580i)11-s + (0.707 + 0.707i)12-s + (−0.212 − 0.977i)14-s + (0.919 + 0.393i)15-s + (0.0475 − 0.998i)16-s + (−0.371 + 0.928i)17-s + (0.415 + 0.909i)18-s + ⋯
L(s)  = 1  + (−0.371 − 0.928i)2-s + (−0.0237 − 0.999i)3-s + (−0.723 + 0.690i)4-s + (−0.415 + 0.909i)5-s + (−0.919 + 0.393i)6-s + (0.986 + 0.165i)7-s + (0.909 + 0.415i)8-s + (−0.998 + 0.0475i)9-s + (0.998 + 0.0475i)10-s + (−0.814 − 0.580i)11-s + (0.707 + 0.707i)12-s + (−0.212 − 0.977i)14-s + (0.919 + 0.393i)15-s + (0.0475 − 0.998i)16-s + (−0.371 + 0.928i)17-s + (0.415 + 0.909i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.200 - 0.979i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.200 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7580514224 - 0.6184308299i\)
\(L(\frac12)\) \(\approx\) \(0.7580514224 - 0.6184308299i\)
\(L(1)\) \(\approx\) \(0.6843172431 - 0.4038934732i\)
\(L(1)\) \(\approx\) \(0.6843172431 - 0.4038934732i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.371 - 0.928i)T \)
3 \( 1 + (-0.0237 - 0.999i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
7 \( 1 + (0.986 + 0.165i)T \)
11 \( 1 + (-0.814 - 0.580i)T \)
17 \( 1 + (-0.371 + 0.928i)T \)
19 \( 1 + (0.672 - 0.739i)T \)
23 \( 1 + (-0.304 - 0.952i)T \)
29 \( 1 + (-0.636 + 0.771i)T \)
31 \( 1 + (0.212 + 0.977i)T \)
37 \( 1 + (0.258 - 0.965i)T \)
41 \( 1 + (0.853 + 0.520i)T \)
43 \( 1 + (0.636 + 0.771i)T \)
47 \( 1 + (-0.959 - 0.281i)T \)
53 \( 1 + (0.281 + 0.959i)T \)
59 \( 1 + (0.999 + 0.0237i)T \)
61 \( 1 + (0.436 - 0.899i)T \)
67 \( 1 + (0.690 - 0.723i)T \)
71 \( 1 + (0.995 + 0.0950i)T \)
73 \( 1 + (0.540 - 0.841i)T \)
79 \( 1 + (-0.540 + 0.841i)T \)
83 \( 1 + (-0.800 - 0.599i)T \)
97 \( 1 + (0.814 - 0.580i)T \)
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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

−21.25647570594859887349066769497, −20.60305829813471077375815559444, −20.161431311014711540120785063855, −19.06032557228947919670851045651, −18.0263187747785207843620914637, −17.42605048843110021730190553457, −16.725509945711459225846831946707, −15.913404932590417898936710315559, −15.50968415929836078059523422983, −14.73167330504646108363696161508, −13.88134076869424622548216044313, −13.11416089562129376961297251645, −11.76637410936462158079523693758, −11.19306390212088580151880553047, −9.961932074587159436851333596528, −9.584460663200842787606084096712, −8.56578990532099549007142693216, −7.93003049617507745369563090281, −7.31789082773160133209863568933, −5.690085121853391212515647316712, −5.247312535419553267435312266940, −4.49724292869630741134843344188, −3.82990868130109139480271710026, −2.115949963870560001953673903657, −0.72740924955778169942809558043, 0.73591207022454790429107342584, 1.93596309603297595607711449403, 2.61730069044262843496461714234, 3.447252961265257156739576558472, 4.65878105543693674325492990052, 5.69481848243570561605014605890, 6.85062797465716079451903111343, 7.74634176398283804175505208467, 8.21010435269832138294595652675, 9.018944536174160277662646154691, 10.41394129554566478834265179044, 11.09533421976923202330822032163, 11.39528618788175491500886274854, 12.46357270741980395179167880093, 13.037180179137323760140930740286, 14.132844450924794906702722037219, 14.44575810157480650508726434004, 15.692542256549036061561015792674, 16.789659657992986645728701387434, 17.88111877633504903595363406985, 18.07037087452411133306524273842, 18.70915628507757149635599469904, 19.57880362108044922045627483113, 20.02478882749689533190991272209, 21.119935672133367574063617211984

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