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Properties

Label 1-920-920.237-r0-0-0
Degree $1$
Conductor $920$
Sign $0.0426 + 0.999i$
Analytic cond. $4.27246$
Root an. cond. $4.27246$
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  + (0.909 + 0.415i)3-s + (−0.540 − 0.841i)7-s + (0.654 + 0.755i)9-s + (−0.142 + 0.989i)11-s + (−0.540 + 0.841i)13-s + (−0.281 + 0.959i)17-s + (0.959 − 0.281i)19-s + (−0.142 − 0.989i)21-s + (0.281 + 0.959i)27-s + (−0.959 − 0.281i)29-s + (0.415 + 0.909i)31-s + (−0.540 + 0.841i)33-s + (0.755 − 0.654i)37-s + (−0.841 + 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)3-s + (−0.540 − 0.841i)7-s + (0.654 + 0.755i)9-s + (−0.142 + 0.989i)11-s + (−0.540 + 0.841i)13-s + (−0.281 + 0.959i)17-s + (0.959 − 0.281i)19-s + (−0.142 − 0.989i)21-s + (0.281 + 0.959i)27-s + (−0.959 − 0.281i)29-s + (0.415 + 0.909i)31-s + (−0.540 + 0.841i)33-s + (0.755 − 0.654i)37-s + (−0.841 + 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0426 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0426 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.0426 + 0.999i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.0426 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.206638121 + 1.156271876i\)
\(L(\frac12)\) \(\approx\) \(1.206638121 + 1.156271876i\)
\(L(1)\) \(\approx\) \(1.229223804 + 0.3778012931i\)
\(L(1)\) \(\approx\) \(1.229223804 + 0.3778012931i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.909 + 0.415i)T \)
7 \( 1 + (-0.540 - 0.841i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (-0.540 + 0.841i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (0.755 - 0.654i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (-0.909 - 0.415i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.540 - 0.841i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (0.989 - 0.142i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.281 + 0.959i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (-0.755 + 0.654i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.755 + 0.654i)T \)
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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

−21.84387534941589681435788468834, −20.61511127490974062529628642064, −20.21357023265997802242810908167, −19.23142560798145115483734836686, −18.58697437900849022805659162104, −18.125622239484781765718056894106, −16.87700992028840827529961879888, −15.90691545945640964646174428971, −15.322084774969261615738752740231, −14.494439285136440532575797369268, −13.560353082096308515677654629848, −13.04650269984083616673145765087, −12.09982710712554896985860690991, −11.366901632165400161189251057343, −9.96936718001082550756199244058, −9.43027195736807236753484598279, −8.52136896096539824095694616876, −7.82008227621043677859292035428, −6.89810774295684888564391394073, −5.89797729170271023602204046950, −5.050757704241318801118297302412, −3.5246482925461586903402531613, −2.97182043832088395486728992627, −2.1077112773449771684916862644, −0.64253734229090847865093737572, 1.48704496013647961608890881802, 2.455217635408365001134746712107, 3.54947111997939213784238725024, 4.265293668220812734799017401356, 5.09141839561667326606708316210, 6.61722016989447735984735997456, 7.28286863519979584611510385759, 8.07347572641918904632500300240, 9.21692870157984697866677016438, 9.785915953028480689105471661726, 10.44327081187680161385785034342, 11.51645203700119420265550973708, 12.69874572414770386544384710464, 13.29189447689282693272181788474, 14.17676657857498769332501139686, 14.82301988243151335494829109810, 15.66188665180905251757523880116, 16.446763984533793737593548185526, 17.199420729001002392119440259694, 18.18620513646441726013401451237, 19.24021235384370902415974359824, 19.77861296720134295330385926384, 20.366905605768177643275154442057, 21.19221087895045572555194912196, 22.01468698412864833087672816111

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