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Properties

Label 1-1157-1157.772-r0-0-0
Degree $1$
Conductor $1157$
Sign $-0.590 + 0.806i$
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  + (−0.989 + 0.142i)2-s + (0.479 + 0.877i)3-s + (0.959 − 0.281i)4-s + (−0.415 − 0.909i)5-s + (−0.599 − 0.800i)6-s + (0.936 + 0.349i)7-s + (−0.909 + 0.415i)8-s + (−0.540 + 0.841i)9-s + (0.540 + 0.841i)10-s + (−0.909 − 0.415i)11-s + (0.707 + 0.707i)12-s + (−0.977 − 0.212i)14-s + (0.599 − 0.800i)15-s + (0.841 − 0.540i)16-s + (−0.989 − 0.142i)17-s + (0.415 − 0.909i)18-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)2-s + (0.479 + 0.877i)3-s + (0.959 − 0.281i)4-s + (−0.415 − 0.909i)5-s + (−0.599 − 0.800i)6-s + (0.936 + 0.349i)7-s + (−0.909 + 0.415i)8-s + (−0.540 + 0.841i)9-s + (0.540 + 0.841i)10-s + (−0.909 − 0.415i)11-s + (0.707 + 0.707i)12-s + (−0.977 − 0.212i)14-s + (0.599 − 0.800i)15-s + (0.841 − 0.540i)16-s + (−0.989 − 0.142i)17-s + (0.415 − 0.909i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.590 + 0.806i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (772, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ -0.590 + 0.806i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3196920367 + 0.6302103525i\)
\(L(\frac12)\) \(\approx\) \(0.3196920367 + 0.6302103525i\)
\(L(1)\) \(\approx\) \(0.6612377410 + 0.2397876227i\)
\(L(1)\) \(\approx\) \(0.6612377410 + 0.2397876227i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.989 + 0.142i)T \)
3 \( 1 + (0.479 + 0.877i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (0.936 + 0.349i)T \)
11 \( 1 + (-0.909 - 0.415i)T \)
17 \( 1 + (-0.989 - 0.142i)T \)
19 \( 1 + (-0.212 - 0.977i)T \)
23 \( 1 + (0.212 + 0.977i)T \)
29 \( 1 + (0.936 + 0.349i)T \)
31 \( 1 + (0.977 + 0.212i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.479 + 0.877i)T \)
43 \( 1 + (-0.936 + 0.349i)T \)
47 \( 1 + (-0.959 + 0.281i)T \)
53 \( 1 + (-0.281 + 0.959i)T \)
59 \( 1 + (-0.877 - 0.479i)T \)
61 \( 1 + (0.0713 - 0.997i)T \)
67 \( 1 + (-0.281 + 0.959i)T \)
71 \( 1 + (-0.415 + 0.909i)T \)
73 \( 1 + (-0.540 - 0.841i)T \)
79 \( 1 + (0.540 + 0.841i)T \)
83 \( 1 + (0.599 + 0.800i)T \)
97 \( 1 + (0.909 - 0.415i)T \)
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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

−20.7404753166708119812997619860, −20.07566551495639904049990492608, −19.31395064301028763032260699644, −18.63613889016102675092465616978, −17.98472307760767294745906394584, −17.64203693432096782637328994460, −16.56121561382170396704552319915, −15.404925942200183194272635124046, −14.93516622695034896702458141000, −14.10917604143122066733720087386, −13.1277070507208010154720039546, −12.13134004124740669596107116070, −11.52737311653450141531661835013, −10.60581211201727533652996741497, −10.11663346171013627863999956309, −8.72609436010846582507038713991, −8.12435289115978037179635576484, −7.57625802230147582263186686093, −6.80491945914212635423470221234, −6.09257790707101465578982667258, −4.49037460608499320401503363209, −3.29759078654080522066369656554, −2.37254468411420461221192708886, −1.807603069140471512234097436387, −0.38143512195871810984739549194, 1.20321206034936552383627572628, 2.36040103767226171602107808790, 3.196917033244384270081901856873, 4.812276610034370033551953448809, 4.91853237019707686116041328711, 6.230860999023132234783572531785, 7.60637786884570041851612200942, 8.24549011848637085975158513979, 8.71500102045855947694989514867, 9.44990308535473704788012097717, 10.37361195375968765062631454709, 11.31543046793360230029840542718, 11.593794227042881505101490410538, 13.00077990187819664816341128636, 13.86419079891466477993342040201, 15.036061778561472580453485506842, 15.55179398002243430092575560382, 15.978428924393761346183433025612, 16.94952872744789295614561115514, 17.576801127042919890667105315599, 18.41823894745119483346020157813, 19.46688497310882624934676010004, 19.95521595979749060382207988267, 20.66341020025624908785102977083, 21.34771462989628639085870422811

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