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Properties

Label 1-1157-1157.20-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.947 + 0.319i$
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  + (−0.971 + 0.235i)2-s + (0.458 − 0.888i)3-s + (0.888 − 0.458i)4-s + (−0.755 + 0.654i)5-s + (−0.235 + 0.971i)6-s + (0.981 − 0.189i)7-s + (−0.755 + 0.654i)8-s + (−0.580 − 0.814i)9-s + (0.580 − 0.814i)10-s + (0.189 − 0.981i)11-s − i·12-s + (−0.909 + 0.415i)14-s + (0.235 + 0.971i)15-s + (0.580 − 0.814i)16-s + (0.235 − 0.971i)17-s + (0.755 + 0.654i)18-s + ⋯
L(s)  = 1  + (−0.971 + 0.235i)2-s + (0.458 − 0.888i)3-s + (0.888 − 0.458i)4-s + (−0.755 + 0.654i)5-s + (−0.235 + 0.971i)6-s + (0.981 − 0.189i)7-s + (−0.755 + 0.654i)8-s + (−0.580 − 0.814i)9-s + (0.580 − 0.814i)10-s + (0.189 − 0.981i)11-s − i·12-s + (−0.909 + 0.415i)14-s + (0.235 + 0.971i)15-s + (0.580 − 0.814i)16-s + (0.235 − 0.971i)17-s + (0.755 + 0.654i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.947 + 0.319i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.947 + 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.432405431 + 0.2346928279i\)
\(L(\frac12)\) \(\approx\) \(1.432405431 + 0.2346928279i\)
\(L(1)\) \(\approx\) \(0.8332217387 - 0.08303368613i\)
\(L(1)\) \(\approx\) \(0.8332217387 - 0.08303368613i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.971 + 0.235i)T \)
3 \( 1 + (0.458 - 0.888i)T \)
5 \( 1 + (-0.755 + 0.654i)T \)
7 \( 1 + (0.981 - 0.189i)T \)
11 \( 1 + (0.189 - 0.981i)T \)
17 \( 1 + (0.235 - 0.971i)T \)
19 \( 1 + (0.580 + 0.814i)T \)
23 \( 1 + (0.0950 + 0.995i)T \)
29 \( 1 + (0.945 + 0.327i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.0475 + 0.998i)T \)
43 \( 1 + (-0.945 + 0.327i)T \)
47 \( 1 + (-0.540 + 0.841i)T \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (0.888 - 0.458i)T \)
61 \( 1 + (-0.371 + 0.928i)T \)
67 \( 1 + (0.458 - 0.888i)T \)
71 \( 1 + (-0.945 + 0.327i)T \)
73 \( 1 + (0.909 - 0.415i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (0.189 + 0.981i)T \)
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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

−20.79112221606002852271982517592, −20.26523926966514071368185359448, −19.67889320569572927852123684414, −18.91580665408593835404709140171, −17.829947125049770566213272581576, −17.17501891582074205227279224004, −16.50248301202236459513600851357, −15.564535696322393319666890690415, −15.17228423935976453415658011759, −14.39236556309105598553406913449, −13.027807110319376495741231491413, −12.09301786541516587546280918298, −11.470258741078088601430578380562, −10.675030008459555263548814479205, −9.86068698905577366980942757815, −9.04487061660261995254697118794, −8.32240056543494617700918368485, −7.89470104687038913624656084385, −6.86768327330038474446364744852, −5.399886165306924721348341381809, −4.49642827976932407266854488194, −3.82572745267010108049929451014, −2.59332738218617820055114329996, −1.721481159020262621550533142016, −0.483101980120425342891193407851, 0.873976212833562376273664144039, 1.47247230422048478679299756448, 2.834485206360039451999401033522, 3.35792553588306432371762353021, 4.99009986146092817644921141554, 6.16346788993207057302884240475, 6.87429945635886783549428582592, 7.85062909308722669312779663188, 7.98295008746979450729307717149, 8.9187406491839995989998831648, 9.95277626103211839995701070093, 10.998006046825595126100167785122, 11.66038495093132980288862625434, 12.03180393666884209646418404752, 13.61803416078723370748791490032, 14.28402656527148982916819335493, 14.81229198006729074188184478550, 15.80347794566391984599328561246, 16.52727381343676193008037193502, 17.60357657007009848180946062803, 18.165137099191773731478988949387, 18.7307434113126826724431707082, 19.42207716082486280890058331940, 20.05770518849756638062352827311, 20.79204990710261110121118062103

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