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Properties

Label 1-5077-5077.1015-r0-0-0
Degree $1$
Conductor $5077$
Sign $0.663 + 0.748i$
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  + (−0.933 + 0.358i)2-s + (−0.656 − 0.753i)3-s + (0.743 − 0.668i)4-s + (0.420 + 0.907i)5-s + (0.883 + 0.468i)6-s + (0.999 − 0.0395i)7-s + (−0.454 + 0.890i)8-s + (−0.136 + 0.990i)9-s + (−0.717 − 0.696i)10-s + (0.941 + 0.337i)11-s + (−0.992 − 0.121i)12-s + (−0.673 + 0.739i)13-s + (−0.918 + 0.394i)14-s + (0.407 − 0.913i)15-s + (0.105 − 0.994i)16-s + (0.859 − 0.511i)17-s + ⋯
L(s)  = 1  + (−0.933 + 0.358i)2-s + (−0.656 − 0.753i)3-s + (0.743 − 0.668i)4-s + (0.420 + 0.907i)5-s + (0.883 + 0.468i)6-s + (0.999 − 0.0395i)7-s + (−0.454 + 0.890i)8-s + (−0.136 + 0.990i)9-s + (−0.717 − 0.696i)10-s + (0.941 + 0.337i)11-s + (−0.992 − 0.121i)12-s + (−0.673 + 0.739i)13-s + (−0.918 + 0.394i)14-s + (0.407 − 0.913i)15-s + (0.105 − 0.994i)16-s + (0.859 − 0.511i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5077\)
Sign: $0.663 + 0.748i$
Analytic conductor: \(23.5774\)
Root analytic conductor: \(23.5774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5077} (1015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5077,\ (0:\ ),\ 0.663 + 0.748i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.118406971 + 0.5031004121i\)
\(L(\frac12)\) \(\approx\) \(1.118406971 + 0.5031004121i\)
\(L(1)\) \(\approx\) \(0.7641083224 + 0.1166940796i\)
\(L(1)\) \(\approx\) \(0.7641083224 + 0.1166940796i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5077 \( 1 \)
good2 \( 1 + (-0.933 + 0.358i)T \)
3 \( 1 + (-0.656 - 0.753i)T \)
5 \( 1 + (0.420 + 0.907i)T \)
7 \( 1 + (0.999 - 0.0395i)T \)
11 \( 1 + (0.941 + 0.337i)T \)
13 \( 1 + (-0.673 + 0.739i)T \)
17 \( 1 + (0.859 - 0.511i)T \)
19 \( 1 + (-0.425 - 0.904i)T \)
23 \( 1 + (0.282 - 0.959i)T \)
29 \( 1 + (0.856 + 0.515i)T \)
31 \( 1 + (-0.00123 + 0.999i)T \)
37 \( 1 + (0.229 + 0.973i)T \)
41 \( 1 + (0.638 - 0.770i)T \)
43 \( 1 + (-0.817 - 0.576i)T \)
47 \( 1 + (0.255 + 0.966i)T \)
53 \( 1 + (0.246 + 0.969i)T \)
59 \( 1 + (-0.821 - 0.570i)T \)
61 \( 1 + (-0.333 - 0.942i)T \)
67 \( 1 + (-0.0210 + 0.999i)T \)
71 \( 1 + (-0.883 - 0.468i)T \)
73 \( 1 + (0.892 + 0.451i)T \)
79 \( 1 + (0.991 - 0.128i)T \)
83 \( 1 + (-0.0655 - 0.997i)T \)
89 \( 1 + (-0.911 - 0.410i)T \)
97 \( 1 + (-0.929 + 0.369i)T \)
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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

−17.83581384361152691576171213460, −17.159354977528761460703020139382, −16.75653204268720632671170356487, −16.43381023254850207544311147158, −15.27890259587509584929759656380, −14.93993754344194431177086976905, −14.03350907639116695876347015098, −12.96320111438459065969622798361, −12.17579079527548691701770871982, −11.8958874216491587780880611162, −11.13347168075766805472034522544, −10.44630558899386155759673227004, −9.71597991628993550594507284627, −9.41311116967888307527961461858, −8.37225597681333083795523543860, −8.11446582247556026005486431771, −7.121094313297639317347558891947, −5.87964151276237774607347882252, −5.782101836202380188433411996507, −4.664821535673918929113589949, −4.0329535483557616155571223031, −3.22847623248479995083119386033, −2.042948071565305095756387554162, −1.29069767085802867025628051093, −0.638606352775388308064525526925, 0.87738370572308497475368089896, 1.59328496630210175132276564152, 2.25458826343492950433545396054, 2.9735521647542721154242662180, 4.60178866834348196580501324400, 5.06831628608701901154248846896, 6.06815587089444092472575883968, 6.68270897901600979068393610217, 7.101704880949881408828077557733, 7.66438217367309725259646415866, 8.58308504782211603717674285923, 9.236093540449863532217053339754, 10.14219713599924029405497286651, 10.71095763115034290584130274647, 11.292564830494038089024725473098, 11.93222343255792112916393614286, 12.41304508574485231290749259303, 13.83416678666915926552458275860, 14.2109389761508400914243679592, 14.68097937229608607591437110265, 15.49615994071285781295471646023, 16.463641681547836943908822187511, 17.15127675879018362212562172294, 17.37557447918043826538557770925, 18.07193967202583742607540401378

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