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Properties

Label 1-5077-5077.1030-r0-0-0
Degree $1$
Conductor $5077$
Sign $-0.131 + 0.991i$
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  + (0.920 − 0.390i)2-s + (−0.892 − 0.451i)3-s + (0.695 − 0.718i)4-s + (0.964 + 0.264i)5-s + (−0.997 − 0.0667i)6-s + (−0.892 + 0.451i)7-s + (0.359 − 0.933i)8-s + (0.593 + 0.805i)9-s + (0.991 − 0.133i)10-s + (−0.944 − 0.328i)11-s + (−0.944 + 0.328i)12-s + (0.920 − 0.390i)13-s + (−0.645 + 0.763i)14-s + (−0.741 − 0.670i)15-s + (−0.0334 − 0.999i)16-s + (0.991 + 0.133i)17-s + ⋯
L(s)  = 1  + (0.920 − 0.390i)2-s + (−0.892 − 0.451i)3-s + (0.695 − 0.718i)4-s + (0.964 + 0.264i)5-s + (−0.997 − 0.0667i)6-s + (−0.892 + 0.451i)7-s + (0.359 − 0.933i)8-s + (0.593 + 0.805i)9-s + (0.991 − 0.133i)10-s + (−0.944 − 0.328i)11-s + (−0.944 + 0.328i)12-s + (0.920 − 0.390i)13-s + (−0.645 + 0.763i)14-s + (−0.741 − 0.670i)15-s + (−0.0334 − 0.999i)16-s + (0.991 + 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5077\)
Sign: $-0.131 + 0.991i$
Analytic conductor: \(23.5774\)
Root analytic conductor: \(23.5774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5077} (1030, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5077,\ (0:\ ),\ -0.131 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3326553617 + 0.3798533967i\)
\(L(\frac12)\) \(\approx\) \(0.3326553617 + 0.3798533967i\)
\(L(1)\) \(\approx\) \(1.141145534 - 0.3610216471i\)
\(L(1)\) \(\approx\) \(1.141145534 - 0.3610216471i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5077 \( 1 \)
good2 \( 1 + (0.920 - 0.390i)T \)
3 \( 1 + (-0.892 - 0.451i)T \)
5 \( 1 + (0.964 + 0.264i)T \)
7 \( 1 + (-0.892 + 0.451i)T \)
11 \( 1 + (-0.944 - 0.328i)T \)
13 \( 1 + (0.920 - 0.390i)T \)
17 \( 1 + (0.991 + 0.133i)T \)
19 \( 1 + (-0.824 - 0.565i)T \)
23 \( 1 + (-0.944 - 0.328i)T \)
29 \( 1 + (-0.892 + 0.451i)T \)
31 \( 1 + (-0.645 + 0.763i)T \)
37 \( 1 + (-0.645 + 0.763i)T \)
41 \( 1 + (0.359 + 0.933i)T \)
43 \( 1 + (-0.824 + 0.565i)T \)
47 \( 1 + (-0.824 + 0.565i)T \)
53 \( 1 + (-0.296 + 0.955i)T \)
59 \( 1 + (0.100 + 0.994i)T \)
61 \( 1 + (-0.979 - 0.199i)T \)
67 \( 1 + (0.593 + 0.805i)T \)
71 \( 1 + (-0.997 - 0.0667i)T \)
73 \( 1 + (-0.166 - 0.986i)T \)
79 \( 1 + (-0.741 + 0.670i)T \)
83 \( 1 + (0.480 - 0.876i)T \)
89 \( 1 + (-0.296 - 0.955i)T \)
97 \( 1 + (-0.420 + 0.907i)T \)
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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

−17.58967077650612418895280197396, −16.882931372920149568227824455784, −16.48398813142004980066472721363, −15.97719099580103710669087101197, −15.31943391266083023111370167597, −14.46584034223183413367610184594, −13.76065943758066702332206998057, −13.07279023084923462307995696583, −12.70874809950189973865598371231, −12.01711114884365003918350820397, −11.11993020516096423148010483718, −10.44100659929065459526636134019, −9.95906323591102671372275180693, −9.19579740452522564023612881889, −8.15672515683220273923564336516, −7.29026460732591904657504503123, −6.565871915400433221887434120351, −5.89677327315238838835428747242, −5.5927568522071170782841319933, −4.82716860927762783913319814576, −3.7891039803573880721051631959, −3.60287940217021701575787486515, −2.27355516608680288438671210526, −1.57684902481731746401367883316, −0.09468280331755777362282112725, 1.25532770102968798330469037776, 1.86173411968010642726864257632, 2.86333949721169337988295630737, 3.25615496426777958313982953108, 4.46751452272837073858272540747, 5.31583262130504299549079302462, 5.829482199018162143619637017778, 6.2046401379911034396285056234, 6.85498930544568533420859617086, 7.747057424814712720061520050742, 8.79089402120064089597616655508, 9.85167759550179525554394405733, 10.3274734140280046349016435132, 10.8296938917772329215290643014, 11.5308488063937131732744525500, 12.47679720149779583245186332903, 12.85760173651748465157422804675, 13.323003743443539808580486372640, 13.91349707416211547008052720802, 14.82871150885859992374464189988, 15.53415726784942305136555393874, 16.37500185927485152413735687843, 16.52998845286725447246147993523, 17.70936595072616160230271240306, 18.4128036840279506518118747338

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