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Properties

Label 1-2873-2873.226-r0-0-0
Degree $1$
Conductor $2873$
Sign $-0.787 + 0.616i$
Analytic cond. $13.3421$
Root an. cond. $13.3421$
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  + (−0.855 − 0.517i)2-s + (−0.963 − 0.268i)3-s + (0.464 + 0.885i)4-s + (−0.977 + 0.209i)5-s + (0.685 + 0.728i)6-s + (0.899 + 0.437i)7-s + (0.0603 − 0.998i)8-s + (0.855 + 0.517i)9-s + (0.945 + 0.326i)10-s + (0.988 − 0.150i)11-s + (−0.209 − 0.977i)12-s + (−0.542 − 0.839i)14-s + (0.998 + 0.0603i)15-s + (−0.568 + 0.822i)16-s + (−0.464 − 0.885i)18-s + (0.707 + 0.707i)19-s + ⋯
L(s)  = 1  + (−0.855 − 0.517i)2-s + (−0.963 − 0.268i)3-s + (0.464 + 0.885i)4-s + (−0.977 + 0.209i)5-s + (0.685 + 0.728i)6-s + (0.899 + 0.437i)7-s + (0.0603 − 0.998i)8-s + (0.855 + 0.517i)9-s + (0.945 + 0.326i)10-s + (0.988 − 0.150i)11-s + (−0.209 − 0.977i)12-s + (−0.542 − 0.839i)14-s + (0.998 + 0.0603i)15-s + (−0.568 + 0.822i)16-s + (−0.464 − 0.885i)18-s + (0.707 + 0.707i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2873 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2873 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2873\)    =    \(13^{2} \cdot 17\)
Sign: $-0.787 + 0.616i$
Analytic conductor: \(13.3421\)
Root analytic conductor: \(13.3421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2873} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2873,\ (0:\ ),\ -0.787 + 0.616i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08873326617 + 0.2573236487i\)
\(L(\frac12)\) \(\approx\) \(0.08873326617 + 0.2573236487i\)
\(L(1)\) \(\approx\) \(0.4838827801 + 0.007218576370i\)
\(L(1)\) \(\approx\) \(0.4838827801 + 0.007218576370i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.855 - 0.517i)T \)
3 \( 1 + (-0.963 - 0.268i)T \)
5 \( 1 + (-0.977 + 0.209i)T \)
7 \( 1 + (0.899 + 0.437i)T \)
11 \( 1 + (0.988 - 0.150i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.382 + 0.923i)T \)
29 \( 1 + (-0.988 - 0.150i)T \)
31 \( 1 + (-0.0302 + 0.999i)T \)
37 \( 1 + (-0.0302 + 0.999i)T \)
41 \( 1 + (-0.871 + 0.491i)T \)
43 \( 1 + (-0.911 + 0.410i)T \)
47 \( 1 + (-0.885 - 0.464i)T \)
53 \( 1 + (0.0603 - 0.998i)T \)
59 \( 1 + (-0.983 - 0.180i)T \)
61 \( 1 + (-0.945 - 0.326i)T \)
67 \( 1 + (-0.464 + 0.885i)T \)
71 \( 1 + (-0.268 + 0.963i)T \)
73 \( 1 + (-0.592 + 0.805i)T \)
79 \( 1 + (-0.0904 + 0.995i)T \)
83 \( 1 + (-0.787 + 0.616i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.542 - 0.839i)T \)
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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

−18.65513171259667250649895489758, −18.0954932238573347309975791164, −17.33514450545696575494868056008, −16.72860991759044109652961630244, −16.42942955959085807621584780923, −15.370726013679711763512271053447, −15.044467002586264853872590353833, −14.27515918028424739732548362389, −13.197608439477218271553259688543, −11.93273787085645507653493172944, −11.75645754701139923465769376138, −10.88773232965218804193242708240, −10.546112655392035431383788088272, −9.32775148823527949570648599903, −8.94678331587053322598980640820, −7.83135599528486230453443687173, −7.33738524636495952355345210137, −6.68009642494122698309510553025, −5.79228315342810667384267124049, −4.836626662829714337138120027501, −4.45423560148083755391057419050, −3.40886970629278248123301214991, −1.816731143231921614167210015071, −1.04245511722722888382520960375, −0.15881247355117074897286028861, 1.39470103180679629034257408086, 1.50088205546069364122506873298, 3.026407594713367038833125060007, 3.782116097935696613219306160185, 4.65822515504573214037887681743, 5.53391781799418558720531302330, 6.63104669553178386440928077961, 7.16768362362648243363983548701, 7.97914300932426602303122823366, 8.48529683325568640374372083311, 9.50637013148342880766763348530, 10.25161628748766017694776178655, 11.24485153519761784733437637390, 11.48653260250897044885419636508, 11.945867189105487631209437546626, 12.64110851469771124827481614711, 13.61682239834911342710124323171, 14.72651000326781544079119800365, 15.40169207999793960212247033562, 16.18925410054945363709001216060, 16.80256845033409261446461250305, 17.38831183038640139556916403871, 18.216187510871766491154160909115, 18.57955310703142064923129309576, 19.28404987967621689136390495903

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