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Properties

Label 1-1157-1157.219-r0-0-0
Degree $1$
Conductor $1157$
Sign $-0.531 - 0.847i$
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  + (−0.814 + 0.580i)2-s + (−0.899 − 0.436i)3-s + (0.327 − 0.945i)4-s + (0.959 − 0.281i)5-s + (0.986 − 0.165i)6-s + (−0.0237 − 0.999i)7-s + (0.281 + 0.959i)8-s + (0.618 + 0.786i)9-s + (−0.618 + 0.786i)10-s + (0.690 + 0.723i)11-s + (−0.707 + 0.707i)12-s + (0.599 + 0.800i)14-s + (−0.986 − 0.165i)15-s + (−0.786 − 0.618i)16-s + (−0.814 − 0.580i)17-s + (−0.959 − 0.281i)18-s + ⋯
L(s)  = 1  + (−0.814 + 0.580i)2-s + (−0.899 − 0.436i)3-s + (0.327 − 0.945i)4-s + (0.959 − 0.281i)5-s + (0.986 − 0.165i)6-s + (−0.0237 − 0.999i)7-s + (0.281 + 0.959i)8-s + (0.618 + 0.786i)9-s + (−0.618 + 0.786i)10-s + (0.690 + 0.723i)11-s + (−0.707 + 0.707i)12-s + (0.599 + 0.800i)14-s + (−0.986 − 0.165i)15-s + (−0.786 − 0.618i)16-s + (−0.814 − 0.580i)17-s + (−0.959 − 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.531 - 0.847i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ -0.531 - 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2362319094 - 0.4270341672i\)
\(L(\frac12)\) \(\approx\) \(0.2362319094 - 0.4270341672i\)
\(L(1)\) \(\approx\) \(0.5752089712 - 0.08454671563i\)
\(L(1)\) \(\approx\) \(0.5752089712 - 0.08454671563i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.814 + 0.580i)T \)
3 \( 1 + (-0.899 - 0.436i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (-0.0237 - 0.999i)T \)
11 \( 1 + (0.690 + 0.723i)T \)
17 \( 1 + (-0.814 - 0.580i)T \)
19 \( 1 + (-0.118 + 0.992i)T \)
23 \( 1 + (-0.919 + 0.393i)T \)
29 \( 1 + (-0.853 + 0.520i)T \)
31 \( 1 + (-0.599 - 0.800i)T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (-0.828 - 0.560i)T \)
43 \( 1 + (0.853 + 0.520i)T \)
47 \( 1 + (-0.654 - 0.755i)T \)
53 \( 1 + (-0.755 - 0.654i)T \)
59 \( 1 + (-0.436 - 0.899i)T \)
61 \( 1 + (0.672 + 0.739i)T \)
67 \( 1 + (0.945 - 0.327i)T \)
71 \( 1 + (-0.235 - 0.971i)T \)
73 \( 1 + (-0.989 - 0.142i)T \)
79 \( 1 + (0.989 + 0.142i)T \)
83 \( 1 + (0.349 - 0.936i)T \)
97 \( 1 + (-0.690 + 0.723i)T \)
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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

−21.80732801109416153735344914207, −20.91075548521734469787950088577, −20.012107675193251677881445204093, −18.950275272388779964626141788662, −18.436528597672953187113750153788, −17.56706107314594860085552992299, −17.26457287947855039132919860739, −16.31894693898262089842747107690, −15.61386744384718186454509043748, −14.7165939727923912598883285380, −13.4764629911350185528164488169, −12.69712515418149392863087525674, −11.87059680372098892128538652557, −11.14733129856324166114690039515, −10.59411094927339110718100119836, −9.58502664708050849452216211026, −9.13842202520975233001840004892, −8.32084219948940444148561027806, −6.77883650661522645751298327072, −6.33266878049349539663129478691, −5.447491099861287443878140074277, −4.30004818384831082094476449588, −3.213185808297943661197718916327, −2.2135570184443877823326599481, −1.29740452825557285886023065960, 0.29715312601398891624990783599, 1.60693825567103266104491976633, 1.948097835968541804393548215579, 4.048537577420302876991382782115, 5.02780784341288800201658130833, 5.82521329039955039778251885679, 6.61344630494607928211468872001, 7.19303038627380066040194090998, 8.0255798464841952647431603267, 9.29146842616606755597142529409, 9.84302250698701898426906672699, 10.60853377286692827293586155139, 11.33575627752519092601008666904, 12.36886496368801566102593421257, 13.30515918655711319507309694061, 14.01918717932294683744407263616, 14.78549233093289401151497421019, 16.10634566340722014423266287203, 16.54078214427306840301525305353, 17.305845509571622199637586669379, 17.71988400583984848585228293708, 18.350227528578847765494148711094, 19.31868488574646303992198519145, 20.20946570911797789443437960558, 20.71808487156249051293652792369

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