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Properties

Label 1-5200-5200.1571-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.998 - 0.0503i$
Analytic cond. $24.1486$
Root an. cond. $24.1486$
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  + (−0.207 + 0.978i)3-s + (−0.866 + 0.5i)7-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (0.978 − 0.207i)17-s + (−0.978 + 0.207i)19-s + (−0.309 − 0.951i)21-s + (−0.913 + 0.406i)23-s + (0.587 − 0.809i)27-s + (0.207 − 0.978i)29-s + (−0.951 − 0.309i)31-s + (0.207 + 0.978i)33-s + (0.104 + 0.994i)37-s + (−0.994 + 0.104i)41-s + (−0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)3-s + (−0.866 + 0.5i)7-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (0.978 − 0.207i)17-s + (−0.978 + 0.207i)19-s + (−0.309 − 0.951i)21-s + (−0.913 + 0.406i)23-s + (0.587 − 0.809i)27-s + (0.207 − 0.978i)29-s + (−0.951 − 0.309i)31-s + (0.207 + 0.978i)33-s + (0.104 + 0.994i)37-s + (−0.994 + 0.104i)41-s + (−0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.998 - 0.0503i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (1571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.998 - 0.0503i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9482007717 + 0.02389878891i\)
\(L(\frac12)\) \(\approx\) \(0.9482007717 + 0.02389878891i\)
\(L(1)\) \(\approx\) \(0.7942817090 + 0.2281215543i\)
\(L(1)\) \(\approx\) \(0.7942817090 + 0.2281215543i\)

Euler product

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

−18.09669688147055322611645719477, −17.230280361559492535951209906367, −16.71134744115806160999920654296, −16.345513552153385332625523595464, −15.23494171836663693420867210192, −14.44917051418660860719087549867, −14.032138836495528137048636211536, −13.21302415533663945579544254638, −12.50496258143729910919673145761, −12.285823192078774672934599726558, −11.38578516247406875479607084276, −10.58512820747317846110731179018, −9.9920184287287115803320767024, −9.110665156795389335739088148630, −8.44812945450216965530252592935, −7.6563596761346764903641102545, −6.830341153955634824860205069702, −6.61187945709654816645018790691, −5.76908255922780631608888709418, −4.98018468334204733710860327081, −3.85028312196157600366186177630, −3.409177885717190702280370516076, −2.28112437241857963131163279997, −1.62970257300948533794241827137, −0.70518351283211150609651999457, 0.35438233362988161956216856825, 1.651675678963276220866763325011, 2.685288438425519804429226847595, 3.50576101625732358322605388727, 3.880693255771305280780186285064, 4.825990769025320157275197647970, 5.67325623893715502182003178401, 6.17522932648167101927442659570, 6.776809812130240045669292653891, 8.016905928112599347222773796546, 8.588790351708769830351715870972, 9.3640355418685707936363366444, 9.896843475728015269140395732978, 10.34388027105503629477222875842, 11.47008436409756754358892672344, 11.75506965740119435427143412521, 12.49999358399825085016433064148, 13.388727373882125156973050510246, 14.07992345747071914990148573190, 14.96484652587855036371175668121, 15.14859847859269785385627030192, 16.25610261151440270902190253200, 16.45687898944701183290674859371, 17.07773327815195295983541656103, 17.87910584034333569280667687747

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