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Properties

Label 1-2600-2600.453-r0-0-0
Degree $1$
Conductor $2600$
Sign $0.999 + 0.0242i$
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  + (−0.207 + 0.978i)3-s + (0.5 + 0.866i)7-s + (−0.913 − 0.406i)9-s + (0.406 + 0.913i)11-s + (−0.207 − 0.978i)17-s + (−0.207 − 0.978i)19-s + (−0.951 + 0.309i)21-s + (−0.406 − 0.913i)23-s + (0.587 − 0.809i)27-s + (−0.978 − 0.207i)29-s + (0.951 + 0.309i)31-s + (−0.978 + 0.207i)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 + (0.5 + 0.866i)7-s + (−0.913 − 0.406i)9-s + (0.406 + 0.913i)11-s + (−0.207 − 0.978i)17-s + (−0.207 − 0.978i)19-s + (−0.951 + 0.309i)21-s + (−0.406 − 0.913i)23-s + (0.587 − 0.809i)27-s + (−0.978 − 0.207i)29-s + (0.951 + 0.309i)31-s + (−0.978 + 0.207i)33-s + (−0.104 − 0.994i)37-s + (0.994 − 0.104i)41-s + (0.866 − 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.999 + 0.0242i$
Analytic conductor: \(12.0743\)
Root analytic conductor: \(12.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (0:\ ),\ 0.999 + 0.0242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329179375 + 0.01609588702i\)
\(L(\frac12)\) \(\approx\) \(1.329179375 + 0.01609588702i\)
\(L(1)\) \(\approx\) \(0.9670187158 + 0.2517079341i\)
\(L(1)\) \(\approx\) \(0.9670187158 + 0.2517079341i\)

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

−19.40974248689948384303511942849, −18.70957340827867530957462573560, −17.910207873338636800940484468893, −17.2256865522027001432328026429, −16.81438376367484473653132093444, −16.02955766274900697398058068629, −14.82109545004470538894513329693, −14.32163015017401027015578360327, −13.534815594889614050915255015532, −13.120942824314764697479292879773, −12.151595219293525789744444166756, −11.49027507618196601466726501226, −10.884961513408304956381419972570, −10.14635587445951451696741899762, −9.03275498091147122095340730500, −8.15422748658923529390865722699, −7.779227830074794734401340341250, −6.90565136162024823176779893760, −6.04493351376076891994281824217, −5.6217912386701127738615478699, −4.329453047502043584798147867895, −3.67526092881650650398400069445, −2.60624558297407797741965213662, −1.49779667263359388050530643427, −1.05295910933096576592954049325, 0.487158166907748692164721734574, 2.08992476075778761491648793546, 2.61814622279583236515176008358, 3.77406187116307418373583652013, 4.60642454979601051838529819347, 5.06411269850948899658905687881, 5.94152430582548930357450517838, 6.782348510079160018616699118922, 7.71755839707853961307787604085, 8.77979534855433576633454289519, 9.155033181892175428676415090809, 9.89306454608117345458004287444, 10.751201344593164452364639907577, 11.46217215128351222800614278300, 12.01564181119883303154440622179, 12.76453010023885707746296283144, 13.8952846265207316770031501693, 14.59611630937090774245201167567, 15.136661580054961636199041764872, 15.79016212266056956519063367034, 16.37301575560385536748646682050, 17.43467919141917722216122460922, 17.70684947896579802584170050036, 18.55043936524704086172833456851, 19.482614295277150451665152006089

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