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Properties

Label 1-1157-1157.383-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.809 - 0.587i$
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  + (0.971 − 0.235i)2-s + (−0.304 − 0.952i)3-s + (0.888 − 0.458i)4-s + (0.654 + 0.755i)5-s + (−0.520 − 0.853i)6-s + (−0.560 + 0.828i)7-s + (0.755 − 0.654i)8-s + (−0.814 + 0.580i)9-s + (0.814 + 0.580i)10-s + (0.189 − 0.981i)11-s + (−0.707 − 0.707i)12-s + (−0.349 + 0.936i)14-s + (0.520 − 0.853i)15-s + (0.580 − 0.814i)16-s + (0.971 + 0.235i)17-s + (−0.654 + 0.755i)18-s + ⋯
L(s)  = 1  + (0.971 − 0.235i)2-s + (−0.304 − 0.952i)3-s + (0.888 − 0.458i)4-s + (0.654 + 0.755i)5-s + (−0.520 − 0.853i)6-s + (−0.560 + 0.828i)7-s + (0.755 − 0.654i)8-s + (−0.814 + 0.580i)9-s + (0.814 + 0.580i)10-s + (0.189 − 0.981i)11-s + (−0.707 − 0.707i)12-s + (−0.349 + 0.936i)14-s + (0.520 − 0.853i)15-s + (0.580 − 0.814i)16-s + (0.971 + 0.235i)17-s + (−0.654 + 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.809 - 0.587i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.809 - 0.587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.770427477 - 0.8999329452i\)
\(L(\frac12)\) \(\approx\) \(2.770427477 - 0.8999329452i\)
\(L(1)\) \(\approx\) \(1.846765606 - 0.4919184631i\)
\(L(1)\) \(\approx\) \(1.846765606 - 0.4919184631i\)

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 \( 1 + (-0.304 - 0.952i)T \)
5 \( 1 + (0.654 + 0.755i)T \)
7 \( 1 + (-0.560 + 0.828i)T \)
11 \( 1 + (0.189 - 0.981i)T \)
17 \( 1 + (0.971 + 0.235i)T \)
19 \( 1 + (-0.165 + 0.986i)T \)
23 \( 1 + (0.771 + 0.636i)T \)
29 \( 1 + (-0.436 - 0.899i)T \)
31 \( 1 + (0.349 - 0.936i)T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 + (0.672 + 0.739i)T \)
43 \( 1 + (0.436 - 0.899i)T \)
47 \( 1 + (0.841 + 0.540i)T \)
53 \( 1 + (-0.540 - 0.841i)T \)
59 \( 1 + (0.952 + 0.304i)T \)
61 \( 1 + (-0.393 - 0.919i)T \)
67 \( 1 + (-0.458 + 0.888i)T \)
71 \( 1 + (0.327 + 0.945i)T \)
73 \( 1 + (0.909 - 0.415i)T \)
79 \( 1 + (-0.909 + 0.415i)T \)
83 \( 1 + (0.479 - 0.877i)T \)
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

−21.31793033085642770324840577095, −20.80042354285205740200806568942, −20.16783962224577212928885631550, −19.534439301023848343687735753451, −17.78648771375632873361555000062, −17.2004510165878889632882614682, −16.52802930930364873199594516761, −16.04716452005670374752470210711, −15.1220466730132278633453503817, −14.32292699234707299976909576738, −13.68749336710978139934669773380, −12.57450990868089261973925876960, −12.35969780022980332873957815656, −11.013824177680186759956635946433, −10.42135257329450943444442122753, −9.535234032314373039704639768327, −8.7981059669817887909421837418, −7.41758677365609992165058223939, −6.67268196160270805486420413428, −5.72283239252705439180418242798, −4.922354074364011514194383537909, −4.400915992540874321590084854815, −3.467327666359779960395495706528, −2.4951073230262495406757642564, −1.05883840881032250393270445950, 1.15953656433752381831124643884, 2.16488983964156382672538663783, 2.91820372557143215700178122009, 3.65089733889334965268070628746, 5.328352724375286748844439908026, 5.9391372796464984198925154118, 6.25889054138157715661463248026, 7.268892633269095283259621136782, 8.17665485041250155326737459050, 9.49314310220654464164188421263, 10.34040817511005288527764783833, 11.32215865309073343192811093920, 11.798417845715031603209873746964, 12.745311695976155977352858666, 13.31112588852141335269161292481, 14.07172661110081947739542597341, 14.68988786282740685483167308113, 15.58597581670310979530510086135, 16.64127923223851994263789060837, 17.2306227699414857065707442702, 18.53966075383693479659070512038, 18.994466268504664990468377442843, 19.28519817478912424404013761520, 20.69617823783155247658752667047, 21.361469483705218314308231548911

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