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Properties

Label 1-1157-1157.148-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.109 - 0.993i$
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  + (−0.909 − 0.415i)2-s + (−0.997 + 0.0713i)3-s + (0.654 + 0.755i)4-s + (0.959 − 0.281i)5-s + (0.936 + 0.349i)6-s + (0.479 − 0.877i)7-s + (−0.281 − 0.959i)8-s + (0.989 − 0.142i)9-s + (−0.989 − 0.142i)10-s + (−0.281 + 0.959i)11-s + (−0.707 − 0.707i)12-s + (−0.800 + 0.599i)14-s + (−0.936 + 0.349i)15-s + (−0.142 + 0.989i)16-s + (−0.909 + 0.415i)17-s + (−0.959 − 0.281i)18-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)2-s + (−0.997 + 0.0713i)3-s + (0.654 + 0.755i)4-s + (0.959 − 0.281i)5-s + (0.936 + 0.349i)6-s + (0.479 − 0.877i)7-s + (−0.281 − 0.959i)8-s + (0.989 − 0.142i)9-s + (−0.989 − 0.142i)10-s + (−0.281 + 0.959i)11-s + (−0.707 − 0.707i)12-s + (−0.800 + 0.599i)14-s + (−0.936 + 0.349i)15-s + (−0.142 + 0.989i)16-s + (−0.909 + 0.415i)17-s + (−0.959 − 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.109 - 0.993i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.109 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6060747282 - 0.5428495780i\)
\(L(\frac12)\) \(\approx\) \(0.6060747282 - 0.5428495780i\)
\(L(1)\) \(\approx\) \(0.6311905040 - 0.2068386514i\)
\(L(1)\) \(\approx\) \(0.6311905040 - 0.2068386514i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.909 - 0.415i)T \)
3 \( 1 + (-0.997 + 0.0713i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (0.479 - 0.877i)T \)
11 \( 1 + (-0.281 + 0.959i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (0.599 - 0.800i)T \)
23 \( 1 + (-0.599 + 0.800i)T \)
29 \( 1 + (0.479 - 0.877i)T \)
31 \( 1 + (0.800 - 0.599i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.997 + 0.0713i)T \)
43 \( 1 + (-0.479 - 0.877i)T \)
47 \( 1 + (-0.654 - 0.755i)T \)
53 \( 1 + (0.755 + 0.654i)T \)
59 \( 1 + (-0.0713 + 0.997i)T \)
61 \( 1 + (-0.212 - 0.977i)T \)
67 \( 1 + (0.755 + 0.654i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (0.989 + 0.142i)T \)
79 \( 1 + (-0.989 - 0.142i)T \)
83 \( 1 + (-0.936 - 0.349i)T \)
97 \( 1 + (0.281 + 0.959i)T \)
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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

−21.348324671040619792065350459019, −20.926985866125532917946430898004, −19.66940104819303431049035154565, −18.598634115011172645025639385645, −18.283829040311745302482957182158, −17.764796156580940320841212615374, −16.94766285215074066506811513567, −16.12851315052725686558188085770, −15.66566583097976648392197257401, −14.500907630252243285134349016637, −13.87590156156046806317071585597, −12.6915651257608079856367701258, −11.74931591307293260364623193323, −11.06872444697536990082103527591, −10.38121804341705682974572750260, −9.62836822815712168086385956771, −8.709317077280589143060918829062, −7.95379246823784473093943602267, −6.692396123895535572146529331819, −6.25866305103055282250408660099, −5.44513693784433866730240991261, −4.86302661808377546747102910866, −2.91933542834811705935088564221, −1.95962785221406303185570577080, −1.05097512806107054601828754448, 0.60185179562575151445815086033, 1.61611525458185173469450397891, 2.34594962240784843314950820484, 3.980710720840187875273115464628, 4.727158822623861818530575478236, 5.78125692932730396190004281400, 6.792503896045382329591577746004, 7.33173223838531937837586690511, 8.39973096079686419883317308826, 9.57482331685853817606591323538, 9.988450843263146634927264190956, 10.73563217426184865056002790175, 11.453555086417616181330437557621, 12.28385726108980258506139696575, 13.15350819916191292427890911917, 13.75531180741787536630075605162, 15.26937938747677218406014797991, 15.88983426863772929158594323542, 16.90954294967393706283134256343, 17.47840219407980106994113676620, 17.70946964900810614090988913973, 18.44619935006278318671040713777, 19.69979435583219839174516377628, 20.28086567578108177610765798596, 21.148305364298791028287221482924

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