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Properties

Label 1-916-916.311-r1-0-0
Degree $1$
Conductor $916$
Sign $0.768 - 0.639i$
Analytic cond. $98.4378$
Root an. cond. $98.4378$
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  + (−0.993 + 0.110i)3-s + (−0.754 + 0.656i)5-s + (0.962 + 0.272i)7-s + (0.975 − 0.218i)9-s + (−0.546 + 0.837i)11-s + (0.945 + 0.324i)13-s + (0.677 − 0.735i)15-s + (0.945 + 0.324i)17-s + (0.191 − 0.981i)19-s + (−0.986 − 0.164i)21-s + (−0.0275 − 0.999i)23-s + (0.137 − 0.990i)25-s + (−0.945 + 0.324i)27-s + (0.716 − 0.697i)29-s + (−0.451 − 0.892i)31-s + ⋯
L(s)  = 1  + (−0.993 + 0.110i)3-s + (−0.754 + 0.656i)5-s + (0.962 + 0.272i)7-s + (0.975 − 0.218i)9-s + (−0.546 + 0.837i)11-s + (0.945 + 0.324i)13-s + (0.677 − 0.735i)15-s + (0.945 + 0.324i)17-s + (0.191 − 0.981i)19-s + (−0.986 − 0.164i)21-s + (−0.0275 − 0.999i)23-s + (0.137 − 0.990i)25-s + (−0.945 + 0.324i)27-s + (0.716 − 0.697i)29-s + (−0.451 − 0.892i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(916\)    =    \(2^{2} \cdot 229\)
Sign: $0.768 - 0.639i$
Analytic conductor: \(98.4378\)
Root analytic conductor: \(98.4378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{916} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 916,\ (1:\ ),\ 0.768 - 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.114610182 - 0.4033470437i\)
\(L(\frac12)\) \(\approx\) \(1.114610182 - 0.4033470437i\)
\(L(1)\) \(\approx\) \(0.8048349350 + 0.07827119452i\)
\(L(1)\) \(\approx\) \(0.8048349350 + 0.07827119452i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (-0.993 + 0.110i)T \)
5 \( 1 + (-0.754 + 0.656i)T \)
7 \( 1 + (0.962 + 0.272i)T \)
11 \( 1 + (-0.546 + 0.837i)T \)
13 \( 1 + (0.945 + 0.324i)T \)
17 \( 1 + (0.945 + 0.324i)T \)
19 \( 1 + (0.191 - 0.981i)T \)
23 \( 1 + (-0.0275 - 0.999i)T \)
29 \( 1 + (0.716 - 0.697i)T \)
31 \( 1 + (-0.451 - 0.892i)T \)
37 \( 1 + (0.350 + 0.936i)T \)
41 \( 1 + (-0.298 - 0.954i)T \)
43 \( 1 + (0.986 + 0.164i)T \)
47 \( 1 + (-0.975 + 0.218i)T \)
53 \( 1 + (-0.401 + 0.915i)T \)
59 \( 1 + (-0.350 + 0.936i)T \)
61 \( 1 + (-0.677 - 0.735i)T \)
67 \( 1 + (0.298 - 0.954i)T \)
71 \( 1 + (0.998 - 0.0550i)T \)
73 \( 1 + (0.137 - 0.990i)T \)
79 \( 1 + (-0.716 - 0.697i)T \)
83 \( 1 + (-0.635 - 0.771i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.926 - 0.376i)T \)
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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

−21.529494085257047014301587714, −21.17105704313662507919099240913, −20.340675631688693730763433162584, −19.322077086010443966635899689522, −18.43275782891590909039212157656, −17.91367579540346828997037538256, −16.92700166735561892214262332609, −16.14887768414338679696793594736, −15.861910485959629030616650988665, −14.595318794921888208077017833675, −13.66911528231984286886355253763, −12.74771118999739470088710759215, −12.0371660496388535940184682941, −11.19538484463884533681343421163, −10.80216910094079447378814686132, −9.6653590433144502063520418321, −8.28963251981508172255731189223, −7.95435422626092956013777230870, −6.94617993711470355432511550641, −5.52125400848353808492403080568, −5.35632291508331793250954581173, −4.15818779624333494229117677147, −3.33779429442363500231395448495, −1.40692123556520509943807491441, −0.94297494044932360381927756626, 0.38443484322081358791370494271, 1.592520434359274461056579583966, 2.82690798891347602825335675556, 4.18276992853036801405920468448, 4.69279779063589957089881123991, 5.78294106006757166823216371183, 6.64724112210976632946919266830, 7.560814774260097603248674374809, 8.22943008154997524913101365827, 9.522056208062303082731419612572, 10.610411720785889794283755831734, 11.00176711609022749253419042953, 11.874922356979556060501042221613, 12.41094642945529584610749013746, 13.57143261934337489659135925944, 14.65416574492049372440914892835, 15.35109787469058056503275635090, 15.91109012240372842031252578076, 16.93861075026539933608752653109, 17.73183579887100368148332688480, 18.47869534844322858857105332334, 18.83439217920994385701478949221, 20.16935883135792798089308755029, 20.97133382503581779037320478654, 21.618740278191160302405496414073

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