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Properties

Label 1-1157-1157.3-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.945 - 0.324i$
Analytic cond. $124.336$
Root an. cond. $124.336$
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  + (−0.995 − 0.0950i)2-s + (−0.560 + 0.828i)3-s + (0.981 + 0.189i)4-s + (0.281 − 0.959i)5-s + (0.636 − 0.771i)6-s + (−0.853 − 0.520i)7-s + (−0.959 − 0.281i)8-s + (−0.371 − 0.928i)9-s + (−0.371 + 0.928i)10-s + (−0.235 + 0.971i)11-s + (−0.707 + 0.707i)12-s + (0.800 + 0.599i)14-s + (0.636 + 0.771i)15-s + (0.928 + 0.371i)16-s + (−0.0950 − 0.995i)17-s + (0.281 + 0.959i)18-s + ⋯
L(s)  = 1  + (−0.995 − 0.0950i)2-s + (−0.560 + 0.828i)3-s + (0.981 + 0.189i)4-s + (0.281 − 0.959i)5-s + (0.636 − 0.771i)6-s + (−0.853 − 0.520i)7-s + (−0.959 − 0.281i)8-s + (−0.371 − 0.928i)9-s + (−0.371 + 0.928i)10-s + (−0.235 + 0.971i)11-s + (−0.707 + 0.707i)12-s + (0.800 + 0.599i)14-s + (0.636 + 0.771i)15-s + (0.928 + 0.371i)16-s + (−0.0950 − 0.995i)17-s + (0.281 + 0.959i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.945 - 0.324i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.945 - 0.324i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7658769347 - 0.1276945144i\)
\(L(\frac12)\) \(\approx\) \(0.7658769347 - 0.1276945144i\)
\(L(1)\) \(\approx\) \(0.5525675986 + 0.002842411233i\)
\(L(1)\) \(\approx\) \(0.5525675986 + 0.002842411233i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.995 - 0.0950i)T \)
3 \( 1 + (-0.560 + 0.828i)T \)
5 \( 1 + (0.281 - 0.959i)T \)
7 \( 1 + (-0.853 - 0.520i)T \)
11 \( 1 + (-0.235 + 0.971i)T \)
17 \( 1 + (-0.0950 - 0.995i)T \)
19 \( 1 + (0.919 + 0.393i)T \)
23 \( 1 + (0.992 - 0.118i)T \)
29 \( 1 + (0.999 + 0.0237i)T \)
31 \( 1 + (-0.599 + 0.800i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (-0.899 - 0.436i)T \)
43 \( 1 + (0.999 - 0.0237i)T \)
47 \( 1 + (-0.755 - 0.654i)T \)
53 \( 1 + (0.755 - 0.654i)T \)
59 \( 1 + (0.560 + 0.828i)T \)
61 \( 1 + (-0.952 + 0.304i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (-0.690 + 0.723i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (-0.989 + 0.142i)T \)
83 \( 1 + (-0.349 - 0.936i)T \)
97 \( 1 + (0.235 + 0.971i)T \)
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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

−21.2467908676419886860203677283, −19.878420894832256964766141809709, −19.16800809409840564652945336, −18.86000811537052528056954948546, −18.12966652117129261123991793911, −17.45163757018267067679279956917, −16.68220264668225695578050861849, −15.8829650732024955747280422209, −15.154163486702000830077233094520, −14.07900123217170631501075945692, −13.23587712316801641156062116784, −12.353061846417541192077753073765, −11.43847366346785354305039048684, −10.88189851370206257017830603826, −10.13333160729382700971089036246, −9.16413721322819428543678183875, −8.29317629079204151870512894401, −7.37824885623260961073400921486, −6.65756235893629999600114131384, −6.05274117593565424770126350826, −5.41426868501195008526688178433, −3.28457458648295518379470748976, −2.71680098081928686134889375398, −1.685304345349869050072114017834, −0.53904089769760815068959357445, 0.45936429820695818486819420301, 1.309039940200887936613063376251, 2.743957002175053301379274875155, 3.70035763216659904951908640165, 4.85547394147853497694276407482, 5.54335629639049210733192007794, 6.729406940774490767416772577777, 7.29898963150609324221600439424, 8.65716420305570933136050655528, 9.23802125969997659898835648893, 10.0389222807937814449853128458, 10.3133239868232441873676488598, 11.55669284710514896871489756481, 12.20733874705639591886758314919, 12.92662035864817719000873965130, 14.099607725867182388713495941749, 15.31352186761949508924200604263, 16.02258775982435022560409265966, 16.36429997986822675781154438491, 17.21756031685299457111568955323, 17.7071976339843224359931240848, 18.57877069400726074919638600641, 19.784948103846680959114585845304, 20.25322003986737369915768415756, 20.83591106806906128268923673633

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