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Properties

Label 1-2675-2675.264-r1-0-0
Degree $1$
Conductor $2675$
Sign $-0.994 + 0.100i$
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  + (0.331 + 0.943i)2-s + (0.513 − 0.858i)3-s + (−0.780 + 0.625i)4-s + (0.979 + 0.200i)6-s + (0.972 + 0.234i)7-s + (−0.848 − 0.528i)8-s + (−0.472 − 0.881i)9-s + (−0.994 − 0.106i)11-s + (0.135 + 0.990i)12-s + (0.0177 + 0.999i)13-s + (0.100 + 0.994i)14-s + (0.217 − 0.976i)16-s + (0.772 − 0.634i)17-s + (0.674 − 0.737i)18-s + (−0.342 + 0.939i)19-s + ⋯
L(s)  = 1  + (0.331 + 0.943i)2-s + (0.513 − 0.858i)3-s + (−0.780 + 0.625i)4-s + (0.979 + 0.200i)6-s + (0.972 + 0.234i)7-s + (−0.848 − 0.528i)8-s + (−0.472 − 0.881i)9-s + (−0.994 − 0.106i)11-s + (0.135 + 0.990i)12-s + (0.0177 + 0.999i)13-s + (0.100 + 0.994i)14-s + (0.217 − 0.976i)16-s + (0.772 − 0.634i)17-s + (0.674 − 0.737i)18-s + (−0.342 + 0.939i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(287.468\)
Root analytic conductor: \(287.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (264, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2675,\ (1:\ ),\ -0.994 + 0.100i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05521144527 + 1.091891318i\)
\(L(\frac12)\) \(\approx\) \(0.05521144527 + 1.091891318i\)
\(L(1)\) \(\approx\) \(1.171476936 + 0.3908661393i\)
\(L(1)\) \(\approx\) \(1.171476936 + 0.3908661393i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 \)
good2 \( 1 + (0.331 + 0.943i)T \)
3 \( 1 + (0.513 - 0.858i)T \)
7 \( 1 + (0.972 + 0.234i)T \)
11 \( 1 + (-0.994 - 0.106i)T \)
13 \( 1 + (0.0177 + 0.999i)T \)
17 \( 1 + (0.772 - 0.634i)T \)
19 \( 1 + (-0.342 + 0.939i)T \)
23 \( 1 + (0.666 - 0.745i)T \)
29 \( 1 + (-0.182 + 0.983i)T \)
31 \( 1 + (-0.815 - 0.578i)T \)
37 \( 1 + (0.0414 - 0.999i)T \)
41 \( 1 + (0.952 + 0.303i)T \)
43 \( 1 + (-0.717 + 0.696i)T \)
47 \( 1 + (-0.00592 + 0.999i)T \)
53 \( 1 + (-0.286 - 0.958i)T \)
59 \( 1 + (0.991 + 0.130i)T \)
61 \( 1 + (0.0769 - 0.997i)T \)
67 \( 1 + (0.240 + 0.970i)T \)
71 \( 1 + (-0.657 + 0.753i)T \)
73 \( 1 + (-0.386 - 0.922i)T \)
79 \( 1 + (-0.905 + 0.424i)T \)
83 \( 1 + (0.451 - 0.892i)T \)
89 \( 1 + (-0.493 - 0.869i)T \)
97 \( 1 + (-0.733 - 0.679i)T \)
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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

−19.05035980242278796397459420263, −18.14606980162181550709641615719, −17.49642463345732138650107025384, −16.80941373707283867948306723430, −15.5334217064651639748284127041, −15.14252223746167525483060711090, −14.61609825684455223924693612971, −13.60613161457012224747093410008, −13.28954160670863905182199342393, −12.34821556942286350461973087947, −11.376187302244874859035362563671, −10.79213017135819980494241768756, −10.32207362283830780426828178308, −9.62727037806377355175922986670, −8.65143459585382766000673467844, −8.157413611251097148226956430529, −7.3419984930224200710857311713, −5.65391762333359070655140427659, −5.281815899479034639941241384784, −4.55523048380578131283798185882, −3.74323346282059926367783324346, −2.97156791871673866580232540756, −2.28098091036687349260870826468, −1.29615637406628524544976251586, −0.14836294318823543286386373894, 1.05076024137355427667127535960, 2.082723136741319835597178276594, 2.93337961560274129925620443892, 3.884978604362240088951419215537, 4.823894089370315114798640316147, 5.55138065248358685878652899253, 6.28263896693983365677677033880, 7.279364029328406703027675332647, 7.63066591854552269935329339088, 8.41365220282582366635658112048, 8.945277182146639531249965862652, 9.820431875792538360386300721549, 11.0866730438006623381517579488, 11.8002502503752451439337566472, 12.71807889137757201898233365365, 13.00468498409167871399803159793, 14.13507303338129876465315674594, 14.455021328214371544604673620908, 14.86277992663385058016603441629, 16.04871003869148472611838550011, 16.50587263575046721973035478491, 17.43592854484242680230671892848, 18.11918042984819582251616950428, 18.63923078400848641681405917398, 19.04295246308065437185567418420

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