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Properties

Label 1-1157-1157.197-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.876 - 0.481i$
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  + (−0.945 + 0.327i)2-s + (−0.118 − 0.992i)3-s + (0.786 − 0.618i)4-s + (−0.841 − 0.540i)5-s + (0.436 + 0.899i)6-s + (0.672 + 0.739i)7-s + (−0.540 + 0.841i)8-s + (−0.971 + 0.235i)9-s + (0.971 + 0.235i)10-s + (0.998 − 0.0475i)11-s + (−0.707 − 0.707i)12-s + (−0.877 − 0.479i)14-s + (−0.436 + 0.899i)15-s + (0.235 − 0.971i)16-s + (−0.945 − 0.327i)17-s + (0.841 − 0.540i)18-s + ⋯
L(s)  = 1  + (−0.945 + 0.327i)2-s + (−0.118 − 0.992i)3-s + (0.786 − 0.618i)4-s + (−0.841 − 0.540i)5-s + (0.436 + 0.899i)6-s + (0.672 + 0.739i)7-s + (−0.540 + 0.841i)8-s + (−0.971 + 0.235i)9-s + (0.971 + 0.235i)10-s + (0.998 − 0.0475i)11-s + (−0.707 − 0.707i)12-s + (−0.877 − 0.479i)14-s + (−0.436 + 0.899i)15-s + (0.235 − 0.971i)16-s + (−0.945 − 0.327i)17-s + (0.841 − 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.876 - 0.481i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7277385917 - 0.1869198639i\)
\(L(\frac12)\) \(\approx\) \(0.7277385917 - 0.1869198639i\)
\(L(1)\) \(\approx\) \(0.6239563122 - 0.1144812191i\)
\(L(1)\) \(\approx\) \(0.6239563122 - 0.1144812191i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.945 + 0.327i)T \)
3 \( 1 + (-0.118 - 0.992i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (0.672 + 0.739i)T \)
11 \( 1 + (0.998 - 0.0475i)T \)
17 \( 1 + (-0.945 - 0.327i)T \)
19 \( 1 + (-0.520 + 0.853i)T \)
23 \( 1 + (-0.999 - 0.0237i)T \)
29 \( 1 + (0.304 - 0.952i)T \)
31 \( 1 + (0.877 + 0.479i)T \)
37 \( 1 + (0.965 - 0.258i)T \)
41 \( 1 + (-0.919 + 0.393i)T \)
43 \( 1 + (-0.304 - 0.952i)T \)
47 \( 1 + (-0.142 - 0.989i)T \)
53 \( 1 + (0.989 + 0.142i)T \)
59 \( 1 + (0.992 + 0.118i)T \)
61 \( 1 + (0.771 + 0.636i)T \)
67 \( 1 + (-0.618 + 0.786i)T \)
71 \( 1 + (0.888 + 0.458i)T \)
73 \( 1 + (0.281 + 0.959i)T \)
79 \( 1 + (-0.281 - 0.959i)T \)
83 \( 1 + (0.997 + 0.0713i)T \)
97 \( 1 + (-0.998 - 0.0475i)T \)
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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

−21.27712207816373724059585146494, −20.230602091208440118986527324323, −19.91039400941128550521967665430, −19.30163027534289751865546977227, −18.04254900147458201606626961757, −17.533321129829468005652211094980, −16.75089191653416935797678684356, −16.079963396165954866796754824129, −15.19798556684915872377441920417, −14.72722949114073997843430405332, −13.66791466220750535510138773272, −12.270706974035779175809610755, −11.41745231623517984405604716990, −11.10649089745095449516181393044, −10.353278698369362249723736219752, −9.541298848590130750561954245761, −8.562360010343112589252766486651, −8.057055230090488259133603477459, −6.90529135730780909991869549553, −6.3342816293580264865572324660, −4.60179516344732336306188688718, −4.07193031871813162413412899634, −3.236634052284192827581658390993, −2.110896942919166522438607970541, −0.678825666693310393362332047603, 0.7266361656769768549158419967, 1.723020955360207473225806780212, 2.47672143249628586676568759840, 4.02468746009268860803181501595, 5.206640149626018203621158577833, 6.11914106219084581769603616810, 6.84427607869620071282816430634, 7.771717190611133804657158178, 8.53261555374562974598005204884, 8.73387029636703336094222528308, 10.017703527467578544062048401097, 11.24734019585214534400518840872, 11.84684719674270709129917916061, 12.10955849963932054515824432544, 13.40707142021183390964614891210, 14.436915989703163712362506317798, 15.07481680055081583974538912606, 15.94657217610125928903587156274, 16.787453421781740593713429715936, 17.42790162043524883865627227176, 18.17687117144185746867269959962, 18.837402395690556781781538298716, 19.55336194611137419803764508266, 20.06798373891130995000427343151, 20.87245461168436873879559942496

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