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Properties

Label 1-1157-1157.532-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.616 - 0.787i$
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  + (−0.841 + 0.540i)2-s + (−0.415 − 0.909i)3-s + (0.415 − 0.909i)4-s + (0.142 + 0.989i)5-s + (0.841 + 0.540i)6-s + (−0.142 − 0.989i)7-s + (0.142 + 0.989i)8-s + (−0.654 + 0.755i)9-s + (−0.654 − 0.755i)10-s + (0.142 − 0.989i)11-s − 12-s + (0.654 + 0.755i)14-s + (0.841 − 0.540i)15-s + (−0.654 − 0.755i)16-s + (0.841 + 0.540i)17-s + (0.142 − 0.989i)18-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (−0.415 − 0.909i)3-s + (0.415 − 0.909i)4-s + (0.142 + 0.989i)5-s + (0.841 + 0.540i)6-s + (−0.142 − 0.989i)7-s + (0.142 + 0.989i)8-s + (−0.654 + 0.755i)9-s + (−0.654 − 0.755i)10-s + (0.142 − 0.989i)11-s − 12-s + (0.654 + 0.755i)14-s + (0.841 − 0.540i)15-s + (−0.654 − 0.755i)16-s + (0.841 + 0.540i)17-s + (0.142 − 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.616 - 0.787i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (532, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.616 - 0.787i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6877647899 - 0.3350466698i\)
\(L(\frac12)\) \(\approx\) \(0.6877647899 - 0.3350466698i\)
\(L(1)\) \(\approx\) \(0.6455738596 - 0.06974632895i\)
\(L(1)\) \(\approx\) \(0.6455738596 - 0.06974632895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.841 + 0.540i)T \)
3 \( 1 + (-0.415 - 0.909i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (-0.142 - 0.989i)T \)
11 \( 1 + (0.142 - 0.989i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.415 + 0.909i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−21.345748285384718742595364823548, −20.69952724602857349583918966980, −19.884274434433823558297626923904, −19.26610884666393594919346038926, −18.00046893489535546479698873585, −17.663286598414674261571713959495, −16.72979516003423061960745990285, −16.22296217572497908674715572337, −15.42475778491292318026811400924, −14.75554219570585279818281665428, −13.160588114586523708219211825915, −12.547120574493429595884762019190, −11.74234327031619089846391651028, −11.28306163073550274683110533191, −9.91625096890257999087014574792, −9.64488492116707025797235074261, −8.916408130823812780930650172610, −8.19758600669969560600348248229, −7.00604230256321162685962513002, −5.88489434176896806360448404096, −4.9891765768237007109133309604, −4.22956675930024182389350227504, −3.08812374995696646007122244548, −2.130280058071637415754188946, −0.910212799754344930957829586561, 0.58012672674578037440963755462, 1.56424015368395491336813140144, 2.65106567746473809621652519307, 3.788313521800750263818881500102, 5.37047578836134523121986418306, 6.19281792252395558576307441048, 6.66185268739073074356866617711, 7.55355190627359562516377885387, 8.042802610933860926177282408776, 9.12413025094176994602356069816, 10.3919038378776754141699956750, 10.702952780356448837705836397714, 11.39299629543053314142800038962, 12.56103900716405335992849758306, 13.55560743196455879951039390751, 14.34808454292447457366576876458, 14.70802564903699875048212768360, 16.11070243792409470314244869424, 16.81240615032964342910551056586, 17.20674667353892474619595691915, 18.179896622590770562070342953649, 18.855263783568805002327594504586, 19.16511148777868333103610785624, 20.0385788525492514390931508675, 21.04725177454440163413043222189

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