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Properties

Label 1-1300-1300.319-r0-0-0
Degree $1$
Conductor $1300$
Sign $-0.981 - 0.192i$
Analytic cond. $6.03717$
Root an. cond. $6.03717$
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  + (−0.978 + 0.207i)3-s + (0.866 + 0.5i)7-s + (0.913 − 0.406i)9-s + (−0.406 + 0.913i)11-s + (−0.978 − 0.207i)17-s + (−0.207 + 0.978i)19-s + (−0.951 − 0.309i)21-s + (−0.913 − 0.406i)23-s + (−0.809 + 0.587i)27-s + (−0.978 + 0.207i)29-s + (0.951 − 0.309i)31-s + (0.207 − 0.978i)33-s + (−0.994 − 0.104i)37-s + (−0.994 − 0.104i)41-s + (0.5 − 0.866i)43-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)3-s + (0.866 + 0.5i)7-s + (0.913 − 0.406i)9-s + (−0.406 + 0.913i)11-s + (−0.978 − 0.207i)17-s + (−0.207 + 0.978i)19-s + (−0.951 − 0.309i)21-s + (−0.913 − 0.406i)23-s + (−0.809 + 0.587i)27-s + (−0.978 + 0.207i)29-s + (0.951 − 0.309i)31-s + (0.207 − 0.978i)33-s + (−0.994 − 0.104i)37-s + (−0.994 − 0.104i)41-s + (0.5 − 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.981 - 0.192i$
Analytic conductor: \(6.03717\)
Root analytic conductor: \(6.03717\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1300,\ (0:\ ),\ -0.981 - 0.192i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02221296355 + 0.2290631246i\)
\(L(\frac12)\) \(\approx\) \(0.02221296355 + 0.2290631246i\)
\(L(1)\) \(\approx\) \(0.6489195608 + 0.1522588102i\)
\(L(1)\) \(\approx\) \(0.6489195608 + 0.1522588102i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.978 + 0.207i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.406 + 0.913i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (-0.207 + 0.978i)T \)
23 \( 1 + (-0.913 - 0.406i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (0.951 - 0.309i)T \)
37 \( 1 + (-0.994 - 0.104i)T \)
41 \( 1 + (-0.994 - 0.104i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.406 + 0.913i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (0.743 + 0.669i)T \)
71 \( 1 + (0.743 - 0.669i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + (0.406 - 0.913i)T \)
97 \( 1 + (0.743 - 0.669i)T \)
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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

−20.7220684659034858826664320042, −19.72926613764891893039414178752, −18.980816212857631923858954380233, −18.072974293499574928955478967650, −17.549793004382974290138722930773, −16.96167633149339420232347417772, −15.99194966257228303213920483974, −15.47077083371714786841464429952, −14.32851214598665820897133098778, −13.45701854363108501189747529305, −12.998675584774952501343525512462, −11.75186981519578822728184463386, −11.29189163981256426005042291485, −10.68132569343054750412833333450, −9.83578653723546369984059536608, −8.59384087910824488229368351271, −7.90348006921603443533144672392, −6.95599873312263545303573618407, −6.23325545477451144117790270537, −5.25768261645167774725611162571, −4.63464925537253895121603330047, −3.67406581746427913738811220220, −2.26408057824035817573072888683, −1.3063824237401214485419344483, −0.10342443409055879936737710289, 1.57865381902649822656978750133, 2.26282445630350050076420919494, 3.84925248152853952082764138862, 4.6373893675515982915629920523, 5.30920261839713285854309337844, 6.14714627257890147281050336278, 7.05170556173509200564269072480, 7.923845393374350060690667934392, 8.84677129759489018301617923936, 9.892241956236839471871066184030, 10.5122273824567439363370694224, 11.353549970404872047736592591, 12.071805667214989185409653234870, 12.6122978361079265277904418171, 13.65804698991929018161616292144, 14.67660809738549231391772390078, 15.39760730275319398624998793801, 15.94061106719919101562499113426, 17.03704186030254703592831685978, 17.50300608934476581368346362057, 18.34360690773248659891385757758, 18.69628340147698510044055943738, 20.0884842727681729713131238794, 20.76450122386296333037683638097, 21.35600438189766498229435175436

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