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Properties

Label 1-920-920.123-r0-0-0
Degree $1$
Conductor $920$
Sign $0.998 + 0.0543i$
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  + (−0.755 − 0.654i)3-s + (−0.909 + 0.415i)7-s + (0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (−0.909 − 0.415i)13-s + (−0.540 − 0.841i)17-s + (−0.841 − 0.540i)19-s + (0.959 + 0.281i)21-s + (0.540 − 0.841i)27-s + (0.841 − 0.540i)29-s + (0.654 + 0.755i)31-s + (0.909 + 0.415i)33-s + (0.989 − 0.142i)37-s + (0.415 + 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)3-s + (−0.909 + 0.415i)7-s + (0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (−0.909 − 0.415i)13-s + (−0.540 − 0.841i)17-s + (−0.841 − 0.540i)19-s + (0.959 + 0.281i)21-s + (0.540 − 0.841i)27-s + (0.841 − 0.540i)29-s + (0.654 + 0.755i)31-s + (0.909 + 0.415i)33-s + (0.989 − 0.142i)37-s + (0.415 + 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.998 + 0.0543i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.998 + 0.0543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6047015980 + 0.01645214523i\)
\(L(\frac12)\) \(\approx\) \(0.6047015980 + 0.01645214523i\)
\(L(1)\) \(\approx\) \(0.6269735730 - 0.07958220798i\)
\(L(1)\) \(\approx\) \(0.6269735730 - 0.07958220798i\)

Euler product

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

−21.8041229930896167106155106227, −21.353951642399393676221891446506, −20.35246070811370139719970382494, −19.52017594757137669807558826161, −18.726646275329686643100748224198, −17.78345080342730806449055329834, −16.850488805668984725045940826035, −16.54569890725457616414435216749, −15.52657220277786939000522986340, −14.99155262367215034696275811425, −13.81991759906273616810217612335, −12.853397925286485875156114959585, −12.291605611789351430607280482056, −11.2147639770226913531801713218, −10.37202335796585906431157110249, −9.9678091933790303971796737317, −8.95234912869965303231203419456, −7.88377142605561245818248218829, −6.6935546766773281911526312059, −6.16697651694312513096386421441, −5.084442531119024945967649375950, −4.26558311561049436457064569974, −3.37539855043456501320976643106, −2.21095257583413925809556293064, −0.47713838217871774340607542841, 0.659474409966243867715864539224, 2.35737231439376704985405761136, 2.77930694196160986015447082680, 4.53325230722659985825541088988, 5.20769721363102363828558923378, 6.206302675268472209743903722022, 6.90925976792935583384400088512, 7.72157867726429018680578489130, 8.72887227326944538675467355791, 9.9062999470109271922902144593, 10.46152641207911092700043396490, 11.57325319298434959749129902871, 12.25527867182378843717780197173, 13.0632517395356043769634884038, 13.46897939497077198302417688688, 14.824220732760731668639378308130, 15.71763134753731950838772876557, 16.28981915857324455027715541461, 17.30819464144572900765626779975, 17.91469573852870639980558829983, 18.67028867519940636831082236532, 19.455429771895507696353610953656, 20.06001601607277154703379877665, 21.359756329929858089615015194154, 21.973538749373506282221770485831

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