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Properties

Label 1-3380-3380.2539-r1-0-0
Degree $1$
Conductor $3380$
Sign $0.145 + 0.989i$
Analytic cond. $363.231$
Root an. cond. $363.231$
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  + (−0.845 − 0.534i)3-s + (−0.692 + 0.721i)7-s + (0.428 + 0.903i)9-s + (0.428 − 0.903i)11-s + (−0.692 + 0.721i)17-s + (−0.5 + 0.866i)19-s + (0.970 − 0.239i)21-s + (−0.5 − 0.866i)23-s + (0.120 − 0.992i)27-s + (−0.996 − 0.0804i)29-s + (0.120 − 0.992i)31-s + (−0.845 + 0.534i)33-s + (−0.919 + 0.391i)37-s + (0.845 + 0.534i)41-s + (−0.919 − 0.391i)43-s + ⋯
L(s)  = 1  + (−0.845 − 0.534i)3-s + (−0.692 + 0.721i)7-s + (0.428 + 0.903i)9-s + (0.428 − 0.903i)11-s + (−0.692 + 0.721i)17-s + (−0.5 + 0.866i)19-s + (0.970 − 0.239i)21-s + (−0.5 − 0.866i)23-s + (0.120 − 0.992i)27-s + (−0.996 − 0.0804i)29-s + (0.120 − 0.992i)31-s + (−0.845 + 0.534i)33-s + (−0.919 + 0.391i)37-s + (0.845 + 0.534i)41-s + (−0.919 − 0.391i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.145 + 0.989i$
Analytic conductor: \(363.231\)
Root analytic conductor: \(363.231\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (2539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3380,\ (1:\ ),\ 0.145 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3368730627 + 0.2910725674i\)
\(L(\frac12)\) \(\approx\) \(0.3368730627 + 0.2910725674i\)
\(L(1)\) \(\approx\) \(0.6493699294 - 0.07416508319i\)
\(L(1)\) \(\approx\) \(0.6493699294 - 0.07416508319i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.845 - 0.534i)T \)
7 \( 1 + (-0.692 + 0.721i)T \)
11 \( 1 + (0.428 - 0.903i)T \)
17 \( 1 + (-0.692 + 0.721i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.996 - 0.0804i)T \)
31 \( 1 + (0.120 - 0.992i)T \)
37 \( 1 + (-0.919 + 0.391i)T \)
41 \( 1 + (0.845 + 0.534i)T \)
43 \( 1 + (-0.919 - 0.391i)T \)
47 \( 1 + (0.354 - 0.935i)T \)
53 \( 1 + (0.970 + 0.239i)T \)
59 \( 1 + (0.948 - 0.316i)T \)
61 \( 1 + (0.278 - 0.960i)T \)
67 \( 1 + (-0.987 + 0.160i)T \)
71 \( 1 + (-0.0402 + 0.999i)T \)
73 \( 1 + (0.568 + 0.822i)T \)
79 \( 1 + (0.354 - 0.935i)T \)
83 \( 1 + (-0.885 - 0.464i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.200 - 0.979i)T \)
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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

−18.159721906003064190779064553348, −17.726467890643968076170220771359, −17.12958200413497074111147283858, −16.435290562835161462944899450917, −15.78288206235222073724983434378, −15.26663068329591495813421081945, −14.40724052212769759678611755106, −13.50850869528180454148411574270, −12.88694106996665087807486722275, −12.1231584413373384694375084480, −11.46332716113667437815904074923, −10.707005237610841714448239736450, −10.14048362830392340200335461727, −9.3524561165035845995261070304, −8.95348557595168072233191898435, −7.45494708862847197222193938325, −6.99546325930566381006901374807, −6.376769242883458385825295904904, −5.448690437528697754550587546550, −4.663707440070840631999278591220, −4.04860196570725452275378210808, −3.3263011805880284242688571383, −2.15613334029164745117538189447, −1.07713268879830911410852273088, −0.13625967570683617564802609237, 0.559870959605665706619533670299, 1.76662367503999732445446975523, 2.36699342626922794454729957552, 3.55810355546770809067623384973, 4.25481637736363781520260476403, 5.39225139828920961759092924927, 6.01067765712022773222258176594, 6.40119991962175672818300653368, 7.21682248239827651228300170992, 8.3273207776407113081029564282, 8.66922988538985215423092838563, 9.806230743517044507602314195621, 10.4255621974714664435910544698, 11.26714337935946178308520146389, 11.8087994825680559095838445667, 12.56151667429820440664359026831, 13.07077101421835895321529120142, 13.7581874285158179286154335972, 14.720594199410890844273115130430, 15.4282175372305520851881075248, 16.281306689875109719475238635727, 16.72203817704855214389710134227, 17.32125007509184258465770369706, 18.295292677675325314572527829325, 18.761033267465926777014470279368

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