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Properties

Label 1-5077-5077.1056-r0-0-0
Degree $1$
Conductor $5077$
Sign $0.975 - 0.218i$
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  + (−0.452 + 0.891i)2-s + (0.373 + 0.927i)3-s + (−0.591 − 0.806i)4-s + (−0.892 + 0.451i)5-s + (−0.996 − 0.0865i)6-s + (−0.930 + 0.365i)7-s + (0.986 − 0.162i)8-s + (−0.721 + 0.692i)9-s + (0.00123 − 0.999i)10-s + (0.0902 + 0.995i)11-s + (0.527 − 0.849i)12-s + (−0.00371 + 0.999i)13-s + (0.0951 − 0.995i)14-s + (−0.751 − 0.659i)15-s + (−0.301 + 0.953i)16-s + (−0.229 + 0.973i)17-s + ⋯
L(s)  = 1  + (−0.452 + 0.891i)2-s + (0.373 + 0.927i)3-s + (−0.591 − 0.806i)4-s + (−0.892 + 0.451i)5-s + (−0.996 − 0.0865i)6-s + (−0.930 + 0.365i)7-s + (0.986 − 0.162i)8-s + (−0.721 + 0.692i)9-s + (0.00123 − 0.999i)10-s + (0.0902 + 0.995i)11-s + (0.527 − 0.849i)12-s + (−0.00371 + 0.999i)13-s + (0.0951 − 0.995i)14-s + (−0.751 − 0.659i)15-s + (−0.301 + 0.953i)16-s + (−0.229 + 0.973i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5077\)
Sign: $0.975 - 0.218i$
Analytic conductor: \(23.5774\)
Root analytic conductor: \(23.5774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5077} (1056, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5077,\ (0:\ ),\ 0.975 - 0.218i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2825103982 + 0.03123894131i\)
\(L(\frac12)\) \(\approx\) \(-0.2825103982 + 0.03123894131i\)
\(L(1)\) \(\approx\) \(0.2944799537 + 0.5313381673i\)
\(L(1)\) \(\approx\) \(0.2944799537 + 0.5313381673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5077 \( 1 \)
good2 \( 1 + (-0.452 + 0.891i)T \)
3 \( 1 + (0.373 + 0.927i)T \)
5 \( 1 + (-0.892 + 0.451i)T \)
7 \( 1 + (-0.930 + 0.365i)T \)
11 \( 1 + (0.0902 + 0.995i)T \)
13 \( 1 + (-0.00371 + 0.999i)T \)
17 \( 1 + (-0.229 + 0.973i)T \)
19 \( 1 + (0.787 + 0.615i)T \)
23 \( 1 + (0.988 - 0.147i)T \)
29 \( 1 + (0.978 + 0.206i)T \)
31 \( 1 + (-0.876 - 0.481i)T \)
37 \( 1 + (-0.963 - 0.266i)T \)
41 \( 1 + (0.241 + 0.970i)T \)
43 \( 1 + (-0.469 - 0.882i)T \)
47 \( 1 + (0.207 - 0.978i)T \)
53 \( 1 + (-0.885 + 0.464i)T \)
59 \( 1 + (0.354 + 0.934i)T \)
61 \( 1 + (-0.999 - 0.0272i)T \)
67 \( 1 + (0.636 - 0.771i)T \)
71 \( 1 + (-0.996 - 0.0865i)T \)
73 \( 1 + (0.100 + 0.994i)T \)
79 \( 1 + (-0.416 - 0.909i)T \)
83 \( 1 + (-0.0680 + 0.997i)T \)
89 \( 1 + (-0.610 + 0.791i)T \)
97 \( 1 + (-0.987 - 0.155i)T \)
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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

−17.53472193082622731581290798161, −16.92613291326988150734556586707, −15.9854453601263312825873300939, −15.750609226495099783040967751050, −14.456506960213561148394126337322, −13.6799908660429743871151912834, −13.25466898943647874380445120096, −12.61188668338310784537841032820, −12.131887480315303789450793501, −11.29619762090558119752519610125, −10.92191268807737008634654114851, −9.84359337535389284873137071881, −9.080030346196357727729444477092, −8.679622427116374645830404425345, −7.89066659696386925631820953639, −7.30062262483341450642637096886, −6.734838792009626258645235852730, −5.54291573225942391586499406182, −4.75349105342163677854217187239, −3.642536762226688561409713949469, −3.01961487410512860082218872526, −2.886767636126297482177215969320, −1.3615551435862788788640507155, −0.74918080249985035427517361340, −0.12214477923257345550515538287, 1.53701926766725510094303041955, 2.57238428703762366542775227484, 3.5547612110676343452945994108, 4.08614395952980571794079088537, 4.79806085872558828293774465494, 5.60505434587455486998941471866, 6.5362157369736077128788299014, 7.03877111140174487566988638834, 7.77687364379146671037189679936, 8.62030174598007333149024118282, 9.08597877689569916279267688014, 9.79617729336255296054921070234, 10.34706562387288804125293832408, 11.01555197579058567805807580832, 11.94211134561503346331202231538, 12.665756687039147030291007149, 13.65253756504859914880476706425, 14.31841352310929662500448176831, 15.084427946590193416967756990679, 15.2263921145830373799348220208, 16.00813098461929122628038706494, 16.50505902366092630583044487957, 17.034054048590121603303015049658, 17.99180097513927759889201612377, 18.79450394718621907741478660262

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