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Properties

Label 1-5077-5077.1045-r0-0-0
Degree $1$
Conductor $5077$
Sign $-0.551 + 0.834i$
Analytic cond. $23.5774$
Root an. cond. $23.5774$
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  + (−0.998 + 0.0618i)2-s + (0.978 + 0.206i)3-s + (0.992 − 0.123i)4-s + (−0.892 − 0.451i)5-s + (−0.989 − 0.145i)6-s + (−0.995 + 0.0914i)7-s + (−0.982 + 0.184i)8-s + (0.914 + 0.404i)9-s + (0.918 + 0.394i)10-s + (−0.205 + 0.978i)11-s + (0.996 + 0.0840i)12-s + (0.345 + 0.938i)13-s + (0.988 − 0.152i)14-s + (−0.780 − 0.625i)15-s + (0.969 − 0.245i)16-s + (0.865 + 0.501i)17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0618i)2-s + (0.978 + 0.206i)3-s + (0.992 − 0.123i)4-s + (−0.892 − 0.451i)5-s + (−0.989 − 0.145i)6-s + (−0.995 + 0.0914i)7-s + (−0.982 + 0.184i)8-s + (0.914 + 0.404i)9-s + (0.918 + 0.394i)10-s + (−0.205 + 0.978i)11-s + (0.996 + 0.0840i)12-s + (0.345 + 0.938i)13-s + (0.988 − 0.152i)14-s + (−0.780 − 0.625i)15-s + (0.969 − 0.245i)16-s + (0.865 + 0.501i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5077\)
Sign: $-0.551 + 0.834i$
Analytic conductor: \(23.5774\)
Root analytic conductor: \(23.5774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5077} (1045, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5077,\ (0:\ ),\ -0.551 + 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4753735392 + 0.8838121928i\)
\(L(\frac12)\) \(\approx\) \(0.4753735392 + 0.8838121928i\)
\(L(1)\) \(\approx\) \(0.7400431494 + 0.2238087871i\)
\(L(1)\) \(\approx\) \(0.7400431494 + 0.2238087871i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5077 \( 1 \)
good2 \( 1 + (-0.998 + 0.0618i)T \)
3 \( 1 + (0.978 + 0.206i)T \)
5 \( 1 + (-0.892 - 0.451i)T \)
7 \( 1 + (-0.995 + 0.0914i)T \)
11 \( 1 + (-0.205 + 0.978i)T \)
13 \( 1 + (0.345 + 0.938i)T \)
17 \( 1 + (0.865 + 0.501i)T \)
19 \( 1 + (-0.660 - 0.750i)T \)
23 \( 1 + (0.0259 + 0.999i)T \)
29 \( 1 + (0.616 - 0.787i)T \)
31 \( 1 + (0.0951 + 0.995i)T \)
37 \( 1 + (0.855 + 0.518i)T \)
41 \( 1 + (-0.127 + 0.991i)T \)
43 \( 1 + (-0.988 + 0.150i)T \)
47 \( 1 + (0.877 - 0.479i)T \)
53 \( 1 + (0.303 - 0.952i)T \)
59 \( 1 + (0.912 + 0.408i)T \)
61 \( 1 + (-0.881 + 0.472i)T \)
67 \( 1 + (0.998 - 0.0494i)T \)
71 \( 1 + (-0.989 - 0.145i)T \)
73 \( 1 + (0.100 - 0.994i)T \)
79 \( 1 + (-0.883 - 0.468i)T \)
83 \( 1 + (-0.943 - 0.332i)T \)
89 \( 1 + (-0.380 + 0.924i)T \)
97 \( 1 + (0.628 + 0.777i)T \)
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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

−18.32892787165399706886500447444, −16.98226177145450415893703048287, −16.432104932960779162654207706479, −15.761249660543811188636520547356, −15.4168858200969604341878020045, −14.54149355030164699748436835529, −13.99613652614012195962472431498, −12.80491287352203655224503615632, −12.594077351087142785243659928220, −11.69957963619825056651184936262, −10.78500777745754939560150506271, −10.28993052007112067443835048597, −9.729597623411517957648494870175, −8.59898978951109638624618533133, −8.50539562082144645691421483077, −7.637839085939263638934114984314, −7.169308235247854526474508509887, −6.33676312634485620405315363249, −5.73093602662235387626178058235, −4.14059751429132233291874766747, −3.451589038521690092409743891783, −2.95681743948265047038576841013, −2.420761504330498610077563959534, −1.04579753032277288536017847977, −0.40096008094386375022358355752, 1.052065817153395003138631241, 1.84232896720255547537380020186, 2.742718652360932376483024341111, 3.41926436732576168647070759501, 4.15120061430504549379970840473, 4.968113188510438566292139640460, 6.18917001384589357754684663352, 6.97447911545181698720751295004, 7.40336670218562918664632084719, 8.26547190712612253532136741215, 8.68711003315334184671689270384, 9.45419528167198806027896495466, 9.894823522561590084213026250239, 10.55624589894786656580335166152, 11.594285076708468986950588577808, 12.11218609112388801665586266699, 12.899722679783730387754903553769, 13.46435173292176925559994525769, 14.60571260835093408611792723973, 15.213887388786386449112279266089, 15.56983798078690118695233036342, 16.324785760531971443928162736201, 16.68435950988966335301096197994, 17.61258973073751074636196256481, 18.504601981539152271155673893820

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