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Properties

Label 1-1157-1157.103-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.677 + 0.735i$
Analytic cond. $124.336$
Root an. cond. $124.336$
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  + (0.654 + 0.755i)2-s + (0.800 + 0.599i)3-s + (−0.142 + 0.989i)4-s + (−0.540 − 0.841i)5-s + (0.0713 + 0.997i)6-s + (0.212 − 0.977i)7-s + (−0.841 + 0.540i)8-s + (0.281 + 0.959i)9-s + (0.281 − 0.959i)10-s + (0.841 + 0.540i)11-s + (−0.707 + 0.707i)12-s + (0.877 − 0.479i)14-s + (0.0713 − 0.997i)15-s + (−0.959 − 0.281i)16-s + (−0.755 − 0.654i)17-s + (−0.540 + 0.841i)18-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s + (0.800 + 0.599i)3-s + (−0.142 + 0.989i)4-s + (−0.540 − 0.841i)5-s + (0.0713 + 0.997i)6-s + (0.212 − 0.977i)7-s + (−0.841 + 0.540i)8-s + (0.281 + 0.959i)9-s + (0.281 − 0.959i)10-s + (0.841 + 0.540i)11-s + (−0.707 + 0.707i)12-s + (0.877 − 0.479i)14-s + (0.0713 − 0.997i)15-s + (−0.959 − 0.281i)16-s + (−0.755 − 0.654i)17-s + (−0.540 + 0.841i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.677 + 0.735i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.677 + 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.709524726 + 1.625074681i\)
\(L(\frac12)\) \(\approx\) \(3.709524726 + 1.625074681i\)
\(L(1)\) \(\approx\) \(1.719009333 + 0.7796109232i\)
\(L(1)\) \(\approx\) \(1.719009333 + 0.7796109232i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.654 + 0.755i)T \)
3 \( 1 + (0.800 + 0.599i)T \)
5 \( 1 + (-0.540 - 0.841i)T \)
7 \( 1 + (0.212 - 0.977i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
17 \( 1 + (-0.755 - 0.654i)T \)
19 \( 1 + (0.877 + 0.479i)T \)
23 \( 1 + (0.479 - 0.877i)T \)
29 \( 1 + (0.977 + 0.212i)T \)
31 \( 1 + (-0.479 - 0.877i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.599 - 0.800i)T \)
43 \( 1 + (0.977 - 0.212i)T \)
47 \( 1 + (0.989 + 0.142i)T \)
53 \( 1 + (0.989 - 0.142i)T \)
59 \( 1 + (0.800 - 0.599i)T \)
61 \( 1 + (0.936 + 0.349i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (0.540 - 0.841i)T \)
73 \( 1 + (-0.959 - 0.281i)T \)
79 \( 1 + (-0.281 + 0.959i)T \)
83 \( 1 + (-0.0713 - 0.997i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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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

−21.060303273022244847701998830106, −19.95225766457787327507975471834, −19.52047520988526690219296261041, −18.97388294917747616999265008781, −18.21766344455272555246575351601, −17.58895036339367407800642611415, −15.759874426093627208645966408, −15.33747002676312744230086043302, −14.56828398017775160950602667836, −13.98355854985050554819154881885, −13.23450653857733746966675637474, −12.21690721025065348049457530028, −11.71565287089356868367121290490, −11.04030794801292221507972845875, −9.91776293918171306559306283263, −8.97875766032600654190970142385, −8.42473215076286224072906688761, −7.11563958930907016457735816422, −6.48128229248198901621347962243, −5.55890610120587496753611878360, −4.29464728533433092519109896821, −3.37481433827109261643517203860, −2.84537773711714750739864947723, −1.91645647805552044661838320033, −0.92689911196244579680898848005, 0.696660975737300423419696083848, 2.17033953102712437723376164159, 3.46127588601915653000405032529, 4.14235586923780579181557939610, 4.613250702352069898484192176329, 5.49485533663977085387064116084, 7.053817417877097415619679344688, 7.35965423298052077153990643364, 8.44240566221322601912789704979, 8.95595421444986505006530085081, 9.86486003909660339744671534057, 11.04552291484964591537977442315, 11.96015936350912246119076624710, 12.7943749404542164990565248681, 13.65677708393694666296791117111, 14.19659516619007393741952688570, 14.9402523318820854282072371136, 15.807268346562066339153793817858, 16.31769041075378516361257290858, 17.01596259251305646667653363655, 17.765435927004446021214137001736, 19.09497684472162682112743619335, 20.0722409721668934214255818762, 20.501808777249972883903124741617, 20.94589515085351669060491026497

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