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Properties

Label 1-1157-1157.202-r0-0-0
Degree $1$
Conductor $1157$
Sign $-0.216 + 0.976i$
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  + (0.458 − 0.888i)2-s + (0.165 + 0.986i)3-s + (−0.580 − 0.814i)4-s + (0.142 + 0.989i)5-s + (0.952 + 0.304i)6-s + (0.393 + 0.919i)7-s + (−0.989 + 0.142i)8-s + (−0.945 + 0.327i)9-s + (0.945 + 0.327i)10-s + (0.371 + 0.928i)11-s + (0.707 − 0.707i)12-s + (0.997 + 0.0713i)14-s + (−0.952 + 0.304i)15-s + (−0.327 + 0.945i)16-s + (0.458 + 0.888i)17-s + (−0.142 + 0.989i)18-s + ⋯
L(s)  = 1  + (0.458 − 0.888i)2-s + (0.165 + 0.986i)3-s + (−0.580 − 0.814i)4-s + (0.142 + 0.989i)5-s + (0.952 + 0.304i)6-s + (0.393 + 0.919i)7-s + (−0.989 + 0.142i)8-s + (−0.945 + 0.327i)9-s + (0.945 + 0.327i)10-s + (0.371 + 0.928i)11-s + (0.707 − 0.707i)12-s + (0.997 + 0.0713i)14-s + (−0.952 + 0.304i)15-s + (−0.327 + 0.945i)16-s + (0.458 + 0.888i)17-s + (−0.142 + 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.216 + 0.976i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ -0.216 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.073502985 + 1.338216953i\)
\(L(\frac12)\) \(\approx\) \(1.073502985 + 1.338216953i\)
\(L(1)\) \(\approx\) \(1.262731607 + 0.3409795528i\)
\(L(1)\) \(\approx\) \(1.262731607 + 0.3409795528i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.458 - 0.888i)T \)
3 \( 1 + (0.165 + 0.986i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (0.393 + 0.919i)T \)
11 \( 1 + (0.371 + 0.928i)T \)
17 \( 1 + (0.458 + 0.888i)T \)
19 \( 1 + (0.899 + 0.436i)T \)
23 \( 1 + (0.828 - 0.560i)T \)
29 \( 1 + (-0.992 - 0.118i)T \)
31 \( 1 + (-0.997 - 0.0713i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (-0.771 - 0.636i)T \)
43 \( 1 + (0.992 - 0.118i)T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (-0.909 + 0.415i)T \)
59 \( 1 + (0.986 + 0.165i)T \)
61 \( 1 + (0.0237 - 0.999i)T \)
67 \( 1 + (0.814 + 0.580i)T \)
71 \( 1 + (0.786 + 0.618i)T \)
73 \( 1 + (0.755 - 0.654i)T \)
79 \( 1 + (-0.755 + 0.654i)T \)
83 \( 1 + (0.212 + 0.977i)T \)
97 \( 1 + (-0.371 + 0.928i)T \)
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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

−20.88397736270862853920439198463, −20.47172165692628293992997635966, −19.51332971456461021536745568361, −18.58839596131594800901591192719, −17.73878974041152064360086255499, −17.088475788195199314026925085289, −16.56406158683390196664855309858, −15.777221571619248838065620350069, −14.51763230215734353020162556547, −13.95507969548299775843482652239, −13.38883175968430570698508320504, −12.79507326715817243296786909424, −11.8003488941327845442332066602, −11.20796151082562023852918718627, −9.46111454798778546722139726238, −8.92511991615282724401586437294, −7.97451798295617072899686443678, −7.42461864336500888168738048834, −6.66362291428493932928178730364, −5.52350496797621553561095175359, −5.10208594541400391662027310953, −3.832964873742851891408661742683, −3.04900555890606481487789464616, −1.45272464107707925855686696320, −0.59445933298776375073270569656, 1.75872145445500449262693002369, 2.450310314651117122835445335328, 3.448513405938072900000145578364, 4.01574875552620561695827191745, 5.27645167875181500838256008044, 5.62684905506638779620914204924, 6.86450702225355041332195987496, 8.16556896021952553469611143102, 9.19853178822089969794561857280, 9.708975409673418290128041920677, 10.58096062661086781680223090531, 11.13230184025186358386350830083, 11.9756603022766235466068558736, 12.69289745444230504174101713305, 13.98336074242813238079209942906, 14.54538206490085318281608217399, 15.061519952583529578771694548887, 15.627240565562144731749127155687, 17.01689060569589658046452278957, 17.790242875593425135722139557992, 18.67976078333843331611352837912, 19.176227142396966896220069851, 20.26730404052606952831137785899, 20.77354976225651116671857791138, 21.57165349915983525209033376616

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