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Properties

Label 1-2600-2600.253-r0-0-0
Degree $1$
Conductor $2600$
Sign $0.994 - 0.101i$
Analytic cond. $12.0743$
Root an. cond. $12.0743$
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 + (−0.5 + 0.866i)7-s + (0.104 + 0.994i)9-s + (0.994 + 0.104i)11-s + (−0.743 + 0.669i)17-s + (0.743 − 0.669i)19-s + (0.951 − 0.309i)21-s + (0.994 + 0.104i)23-s + (0.587 − 0.809i)27-s + (0.669 − 0.743i)29-s + (−0.951 − 0.309i)31-s + (−0.669 − 0.743i)33-s + (−0.913 − 0.406i)37-s + (0.406 − 0.913i)41-s + (−0.866 − 0.5i)43-s + ⋯
L(s)  = 1  + (−0.743 − 0.669i)3-s + (−0.5 + 0.866i)7-s + (0.104 + 0.994i)9-s + (0.994 + 0.104i)11-s + (−0.743 + 0.669i)17-s + (0.743 − 0.669i)19-s + (0.951 − 0.309i)21-s + (0.994 + 0.104i)23-s + (0.587 − 0.809i)27-s + (0.669 − 0.743i)29-s + (−0.951 − 0.309i)31-s + (−0.669 − 0.743i)33-s + (−0.913 − 0.406i)37-s + (0.406 − 0.913i)41-s + (−0.866 − 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.994 - 0.101i$
Analytic conductor: \(12.0743\)
Root analytic conductor: \(12.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (0:\ ),\ 0.994 - 0.101i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.179767485 - 0.05989248580i\)
\(L(\frac12)\) \(\approx\) \(1.179767485 - 0.05989248580i\)
\(L(1)\) \(\approx\) \(0.8579680523 - 0.06578872407i\)
\(L(1)\) \(\approx\) \(0.8579680523 - 0.06578872407i\)

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 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.994 + 0.104i)T \)
17 \( 1 + (-0.743 + 0.669i)T \)
19 \( 1 + (0.743 - 0.669i)T \)
23 \( 1 + (0.994 + 0.104i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (-0.951 - 0.309i)T \)
37 \( 1 + (-0.913 - 0.406i)T \)
41 \( 1 + (0.406 - 0.913i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (0.994 - 0.104i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (-0.978 + 0.207i)T \)
71 \( 1 + (-0.207 + 0.978i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (0.994 + 0.104i)T \)
97 \( 1 + (0.978 + 0.207i)T \)
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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

−19.640752506391988712262013204553, −18.4694113201171723156685863483, −17.91015759537155368123460443484, −17.03786928997388072517827496252, −16.574569038753837971870821111529, −16.08192120587240322124524486294, −15.19108038567440858260167463100, −14.44880361920209322855226103192, −13.722090455795612330187377409994, −12.90014852676405654357564000012, −12.02874826813671780230754264309, −11.45957389718679970580223891524, −10.67127813802918198820452340292, −10.09458910368964605361520344418, −9.28287564518225171475308259321, −8.76888773041630856409594688751, −7.38903017661791688012019695772, −6.79633408477858704006730723807, −6.19172902688941344599520457859, −5.13976665915501283627504807022, −4.54693369731152851232275101354, −3.61187106479917592758718031578, −3.14611952394922638970988523095, −1.52453533577864993308415347882, −0.67183541809500114125372313531, 0.704473676822912931913838363834, 1.76741371822229991558160449203, 2.50910856162457495960661893267, 3.578098889868115920388994205330, 4.5928051869679636254935287666, 5.4668204992597506124986431543, 6.08222641665498773165001126820, 6.85089413510955543611432339714, 7.349168389931059670828252646149, 8.60439242975545762143888979052, 9.03396090405678029369021282163, 9.987971253280737495349542060547, 10.911761187901958100437915729974, 11.59243645246769780540952333074, 12.132128787345023839162103399196, 12.86681644701985991228673974243, 13.44141112942519011928410438100, 14.31545548540722795704507732072, 15.25634500277718760426646920037, 15.81114795268645938302087355224, 16.67118905957278477562276339127, 17.33579643006679003778748974245, 17.877237541254977518369553402234, 18.63355636752957758933533104387, 19.4591369748458439971966647438

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