Properties

Label 1-2675-2675.234-r1-0-0
Degree $1$
Conductor $2675$
Sign $-0.389 + 0.920i$
Analytic cond. $287.468$
Root an. cond. $287.468$
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  + (0.523 + 0.851i)2-s + (−0.0533 + 0.998i)3-s + (−0.451 + 0.892i)4-s + (−0.878 + 0.477i)6-s + (−0.717 + 0.696i)7-s + (−0.996 + 0.0828i)8-s + (−0.994 − 0.106i)9-s + (−0.683 + 0.729i)11-s + (−0.867 − 0.498i)12-s + (0.135 − 0.990i)13-s + (−0.969 − 0.246i)14-s + (−0.592 − 0.805i)16-s + (−0.999 − 0.0355i)17-s + (−0.430 − 0.902i)18-s + (−0.553 − 0.832i)19-s + ⋯
L(s)  = 1  + (0.523 + 0.851i)2-s + (−0.0533 + 0.998i)3-s + (−0.451 + 0.892i)4-s + (−0.878 + 0.477i)6-s + (−0.717 + 0.696i)7-s + (−0.996 + 0.0828i)8-s + (−0.994 − 0.106i)9-s + (−0.683 + 0.729i)11-s + (−0.867 − 0.498i)12-s + (0.135 − 0.990i)13-s + (−0.969 − 0.246i)14-s + (−0.592 − 0.805i)16-s + (−0.999 − 0.0355i)17-s + (−0.430 − 0.902i)18-s + (−0.553 − 0.832i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $-0.389 + 0.920i$
Analytic conductor: \(287.468\)
Root analytic conductor: \(287.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2675,\ (1:\ ),\ -0.389 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4499758076 + 0.6791995871i\)
\(L(\frac12)\) \(\approx\) \(0.4499758076 + 0.6791995871i\)
\(L(1)\) \(\approx\) \(0.5226185725 + 0.6824055462i\)
\(L(1)\) \(\approx\) \(0.5226185725 + 0.6824055462i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 \)
good2 \( 1 + (0.523 + 0.851i)T \)
3 \( 1 + (-0.0533 + 0.998i)T \)
7 \( 1 + (-0.717 + 0.696i)T \)
11 \( 1 + (-0.683 + 0.729i)T \)
13 \( 1 + (0.135 - 0.990i)T \)
17 \( 1 + (-0.999 - 0.0355i)T \)
19 \( 1 + (-0.553 - 0.832i)T \)
23 \( 1 + (-0.639 + 0.768i)T \)
29 \( 1 + (0.353 + 0.935i)T \)
31 \( 1 + (0.872 + 0.487i)T \)
37 \( 1 + (0.666 - 0.745i)T \)
41 \( 1 + (-0.252 + 0.967i)T \)
43 \( 1 + (0.147 - 0.989i)T \)
47 \( 1 + (-0.842 + 0.538i)T \)
53 \( 1 + (0.924 - 0.381i)T \)
59 \( 1 + (-0.998 - 0.0474i)T \)
61 \( 1 + (0.440 - 0.897i)T \)
67 \( 1 + (-0.229 - 0.973i)T \)
71 \( 1 + (-0.966 - 0.257i)T \)
73 \( 1 + (0.910 + 0.413i)T \)
79 \( 1 + (-0.297 - 0.954i)T \)
83 \( 1 + (-0.563 + 0.826i)T \)
89 \( 1 + (0.725 - 0.687i)T \)
97 \( 1 + (-0.848 - 0.528i)T \)
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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

−19.0766848350021720628613117455, −18.49876728479312154733517570490, −17.7504488477245931240571473536, −16.823072808004784990530623128829, −16.187080632506177489033770102988, −15.16618299915420659888805461558, −14.26094414674504274958144368178, −13.53404620596683675479865296854, −13.3937628235999799811061632337, −12.5256073970208059131976668245, −11.826581663881622117416981543923, −11.16486119276775649454057947539, −10.43454321436618920216691659486, −9.7253723836576852379848542021, −8.6799118578873300415086507383, −8.11857240549340922186775907939, −6.92951886929287274570267853801, −6.27517321411469315283222049855, −5.83651919039192825602549072806, −4.52229715667810970675977196214, −3.96426985637957754773563598156, −2.86297803344668824295568152175, −2.31855278539883894562530381493, −1.34651946012213967597864590967, −0.415308106108514609066742627659, 0.23508430216066095296355726388, 2.37439467974938367633188860665, 2.98041416053151762340505517951, 3.77937384777537724482833050325, 4.76384361625621924304988269194, 5.16318860376031381427434780723, 6.053001712909143480639220919036, 6.65077495414542460791044579209, 7.70451145809025137497170213224, 8.50349394137602135503904627093, 9.11831819699978135765279465204, 9.84762631708786741337236933957, 10.63749319641007805462857254501, 11.55202087108050646835797342683, 12.42400537443051796388007937008, 13.023077255930799776228078334815, 13.69591106131409858450597592806, 14.7296337509886899193328895061, 15.314441451562909459626906402497, 15.7018015312346832180734499626, 16.15077976756634762969137221554, 17.150292319877861513419677124204, 17.81891297798055563229452091582, 18.220862451243081032974427917449, 19.61863856178300161529680344831

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