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Properties

Label 1-1157-1157.467-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.998 - 0.0500i$
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  + (−0.415 − 0.909i)2-s + (0.654 + 0.755i)3-s + (−0.654 + 0.755i)4-s + (0.959 + 0.281i)5-s + (0.415 − 0.909i)6-s + (−0.959 − 0.281i)7-s + (0.959 + 0.281i)8-s + (−0.142 + 0.989i)9-s + (−0.142 − 0.989i)10-s + (0.959 − 0.281i)11-s − 12-s + (0.142 + 0.989i)14-s + (0.415 + 0.909i)15-s + (−0.142 − 0.989i)16-s + (0.415 − 0.909i)17-s + (0.959 − 0.281i)18-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)2-s + (0.654 + 0.755i)3-s + (−0.654 + 0.755i)4-s + (0.959 + 0.281i)5-s + (0.415 − 0.909i)6-s + (−0.959 − 0.281i)7-s + (0.959 + 0.281i)8-s + (−0.142 + 0.989i)9-s + (−0.142 − 0.989i)10-s + (0.959 − 0.281i)11-s − 12-s + (0.142 + 0.989i)14-s + (0.415 + 0.909i)15-s + (−0.142 − 0.989i)16-s + (0.415 − 0.909i)17-s + (0.959 − 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.998 - 0.0500i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.998 - 0.0500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.715942472 - 0.04296098593i\)
\(L(\frac12)\) \(\approx\) \(1.715942472 - 0.04296098593i\)
\(L(1)\) \(\approx\) \(1.186087589 - 0.1062876823i\)
\(L(1)\) \(\approx\) \(1.186087589 - 0.1062876823i\)

Euler product

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

−21.454465609749194251598231994236, −20.09716128471680071636321764447, −19.57381993654291050869781853935, −18.9825406674612788552508027099, −18.07691815273774129875593096339, −17.35695649670456382510051221225, −16.94553764064151704755663904255, −15.75957820480681206088245203945, −15.133746011553516935157145110699, −14.120167641332322531698912344137, −13.76673585342017432449357200633, −12.81181506963171582218254865450, −12.32178764759602672274092554397, −10.76853165441557754990151069964, −9.590751869361040810808376849338, −9.344700183518085184635036871571, −8.603734671268195926644135528105, −7.618605961950352414157400096170, −6.55617254598960390026747283793, −6.37134129240065376687972304949, −5.37272960323456444786202512283, −4.11915518944830689294247290514, −2.95160867199019257195581676209, −1.78848386221090735025206550159, −0.96173974224157596254877232905, 1.04073631716616317744844130181, 2.289708017955733150863280853317, 2.97445687464949982905939699153, 3.74937754324940707254532203840, 4.61280865708772439139344067081, 5.82167312590481411323479982962, 6.84777366227866027116019284229, 7.9607095246650423458605214195, 8.98401353090961785479182943634, 9.46834013121534818298351146109, 10.11996220502530246015645641633, 10.662809732159198306733863613274, 11.72598528600695369198616484805, 12.70152757277430915995482127892, 13.538775130128817245851073977305, 14.10450240174375895947343772588, 14.77887827112916128444007307055, 16.2533356163236196982584099680, 16.614494724337827629514298536763, 17.35385527929500180279666875266, 18.635339359112868056940443110871, 18.871636472780230263457049548689, 20.00523656051412947898956211466, 20.34803345598292810656649812954, 21.25332353295805941935943377780

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