Properties

Label 1-2675-2675.1589-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: $\chi_{2675} (1589, \cdot )$
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

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

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