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Properties

Label 1-5077-5077.108-r0-0-0
Degree $1$
Conductor $5077$
Sign $0.979 + 0.200i$
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  + (−0.881 − 0.472i)2-s + (0.994 + 0.103i)3-s + (0.552 + 0.833i)4-s + (0.231 − 0.972i)5-s + (−0.827 − 0.561i)6-s + (−0.888 − 0.459i)7-s + (−0.0927 − 0.995i)8-s + (0.978 + 0.206i)9-s + (−0.664 + 0.747i)10-s + (−0.357 + 0.934i)11-s + (0.463 + 0.886i)12-s + (0.820 + 0.572i)13-s + (0.564 + 0.825i)14-s + (0.331 − 0.943i)15-s + (−0.389 + 0.921i)16-s + (−0.707 + 0.706i)17-s + ⋯
L(s)  = 1  + (−0.881 − 0.472i)2-s + (0.994 + 0.103i)3-s + (0.552 + 0.833i)4-s + (0.231 − 0.972i)5-s + (−0.827 − 0.561i)6-s + (−0.888 − 0.459i)7-s + (−0.0927 − 0.995i)8-s + (0.978 + 0.206i)9-s + (−0.664 + 0.747i)10-s + (−0.357 + 0.934i)11-s + (0.463 + 0.886i)12-s + (0.820 + 0.572i)13-s + (0.564 + 0.825i)14-s + (0.331 − 0.943i)15-s + (−0.389 + 0.921i)16-s + (−0.707 + 0.706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5077\)
Sign: $0.979 + 0.200i$
Analytic conductor: \(23.5774\)
Root analytic conductor: \(23.5774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5077} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5077,\ (0:\ ),\ 0.979 + 0.200i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.495191388 + 0.1511306918i\)
\(L(\frac12)\) \(\approx\) \(1.495191388 + 0.1511306918i\)
\(L(1)\) \(\approx\) \(0.9692848863 - 0.1581863555i\)
\(L(1)\) \(\approx\) \(0.9692848863 - 0.1581863555i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5077 \( 1 \)
good2 \( 1 + (-0.881 - 0.472i)T \)
3 \( 1 + (0.994 + 0.103i)T \)
5 \( 1 + (0.231 - 0.972i)T \)
7 \( 1 + (-0.888 - 0.459i)T \)
11 \( 1 + (-0.357 + 0.934i)T \)
13 \( 1 + (0.820 + 0.572i)T \)
17 \( 1 + (-0.707 + 0.706i)T \)
19 \( 1 + (0.995 - 0.0988i)T \)
23 \( 1 + (0.716 + 0.697i)T \)
29 \( 1 + (-0.899 + 0.437i)T \)
31 \( 1 + (-0.952 + 0.304i)T \)
37 \( 1 + (-0.248 - 0.968i)T \)
41 \( 1 + (0.980 - 0.196i)T \)
43 \( 1 + (-0.901 - 0.433i)T \)
47 \( 1 + (0.270 - 0.962i)T \)
53 \( 1 + (-0.807 + 0.590i)T \)
59 \( 1 + (0.669 - 0.742i)T \)
61 \( 1 + (0.961 + 0.273i)T \)
67 \( 1 + (-0.521 - 0.853i)T \)
71 \( 1 + (-0.827 - 0.561i)T \)
73 \( 1 + (-0.741 + 0.670i)T \)
79 \( 1 + (0.719 + 0.694i)T \)
83 \( 1 + (0.769 + 0.638i)T \)
89 \( 1 + (0.556 + 0.830i)T \)
97 \( 1 + (0.902 + 0.431i)T \)
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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

−18.24916573512573321074558167737, −17.60869546749114724567630006098, −16.3689029766031322646327788297, −16.02531249711771584873799845758, −15.42927673695939669276724918670, −14.78590845205433816854860753875, −14.20981299860893358122964008046, −13.26908780158230152631523096167, −13.15861548750407554266993154340, −11.67455067858212316662832196713, −11.11614103804220690045411684668, −10.33583815988018499149589645097, −9.77750722023436271356104942149, −9.05873202855413626636103911315, −8.65235097952291674736092362487, −7.69069899584269289446442042640, −7.277151482836707993785678356751, −6.37277790779878686707152118516, −6.0136896898759008589882981380, −5.071050015111294396103196566725, −3.66030251004114193590775319, −2.96082020491312700873544805942, −2.632242157299822387756958183838, −1.618684081464667221478811867959, −0.525876527236010714932119726715, 0.906009610576930043496704214770, 1.74683462449822032840969715836, 2.19980203570737990000912306426, 3.42755833863629231469540397952, 3.74336939529311781857798699074, 4.5861084641627668509472301551, 5.645059881486486857597208522326, 6.804446800423843891474117768, 7.2640368217430446914726064203, 7.96516804456305392257078826056, 8.87921993880086937149372404486, 9.202527819075169819000388348091, 9.66760242676924001804710660492, 10.45377467551496367394923901050, 11.102924631399650327338724074482, 12.16972448719794296838795592229, 12.822332512226811501364452925284, 13.20193817627210865227516646222, 13.749322819403863196307170431633, 14.874150217563018841554566505018, 15.70690428327885873094331756345, 16.06382680687645660063613767558, 16.67640909330372240891107581221, 17.47399231155888456910135311667, 18.10738909687595568757375091031

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