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Properties

Label 1-2600-2600.933-r1-0-0
Degree $1$
Conductor $2600$
Sign $-0.889 + 0.456i$
Analytic cond. $279.408$
Root an. cond. $279.408$
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  + (0.743 − 0.669i)3-s + (0.866 − 0.5i)7-s + (0.104 − 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.743 − 0.669i)17-s + (−0.669 + 0.743i)19-s + (0.309 − 0.951i)21-s + (0.994 − 0.104i)23-s + (−0.587 − 0.809i)27-s + (0.669 + 0.743i)29-s + (−0.309 − 0.951i)31-s + (−0.743 − 0.669i)33-s + (−0.406 − 0.913i)37-s + (−0.913 + 0.406i)41-s + (−0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)3-s + (0.866 − 0.5i)7-s + (0.104 − 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.743 − 0.669i)17-s + (−0.669 + 0.743i)19-s + (0.309 − 0.951i)21-s + (0.994 − 0.104i)23-s + (−0.587 − 0.809i)27-s + (0.669 + 0.743i)29-s + (−0.309 − 0.951i)31-s + (−0.743 − 0.669i)33-s + (−0.406 − 0.913i)37-s + (−0.913 + 0.406i)41-s + (−0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $-0.889 + 0.456i$
Analytic conductor: \(279.408\)
Root analytic conductor: \(279.408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (933, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (1:\ ),\ -0.889 + 0.456i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3865467907 - 1.601783569i\)
\(L(\frac12)\) \(\approx\) \(-0.3865467907 - 1.601783569i\)
\(L(1)\) \(\approx\) \(1.143885372 - 0.6193713986i\)
\(L(1)\) \(\approx\) \(1.143885372 - 0.6193713986i\)

Euler product

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

−19.73310016250952865041809633115, −18.99840878433606604084912139846, −18.20342395696931800892269214101, −17.34427836556466756229622802623, −16.94985472246233396482287412623, −15.660244206672925335467706495584, −15.23610156420071173215117592564, −14.92933583061810943441320817837, −13.9502132442654758917626294419, −13.332215873589299238139588933329, −12.459692954867220507253227119411, −11.66619982974124586427403303239, −10.70425433446429607633622648455, −10.35917940886365798582751563136, −9.23750074691226151405202307645, −8.7752332355187633612977635814, −8.12022452144009274828250505755, −7.25690803721607743722120795286, −6.43309120574289885725148039245, −5.11152817065592988018117911798, −4.7864199083960436158456971734, −3.99448620759539881032412920483, −2.926234738720427968329677566453, −2.174748161901258489768667952881, −1.49288294118061903691867930560, 0.219308113390128734364647827, 1.12518761335893224735773751729, 1.92671162880865344235211058801, 2.83768756660738327902266387452, 3.63963864632815052802650181, 4.506497441146188471574791924287, 5.43136968449027404069972616610, 6.44752841360500067042975181003, 7.07578779611243573523250997511, 7.91486407409722405289839898004, 8.48052393142435273876301745830, 9.0750169304788487976879848375, 10.07844698346764213227437071958, 11.05379904969000334127857293847, 11.448256937984922781714217391657, 12.53683015535733789272161157498, 13.1196951186737832474847447478, 13.952961250968365005906043995048, 14.27644881361636248807213091001, 15.085573226867659255686564238378, 15.84811729155485680169811184485, 16.85751046283756766042382918726, 17.35860337790756527386097714193, 18.437950868588568376255353073434, 18.55331703938347615246543787567

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