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Properties

Label 2-605-5.4-c1-0-18
Degree $2$
Conductor $605$
Sign $-0.579 - 0.815i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.23i·2-s + 0.363i·3-s + 0.477·4-s + (1.29 + 1.82i)5-s − 0.449·6-s + 2.58i·7-s + 3.05i·8-s + 2.86·9-s + (−2.24 + 1.59i)10-s + 0.173i·12-s − 2.75i·13-s − 3.19·14-s + (−0.663 + 0.471i)15-s − 2.81·16-s − 3.85i·17-s + 3.53i·18-s + ⋯
L(s)  = 1  + 0.872i·2-s + 0.210i·3-s + 0.238·4-s + (0.579 + 0.815i)5-s − 0.183·6-s + 0.977i·7-s + 1.08i·8-s + 0.955·9-s + (−0.711 + 0.505i)10-s + 0.0501i·12-s − 0.765i·13-s − 0.852·14-s + (−0.171 + 0.121i)15-s − 0.704·16-s − 0.934i·17-s + 0.834i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.579 - 0.815i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.579 - 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.894709 + 1.73306i\)
\(L(\frac12)\) \(\approx\) \(0.894709 + 1.73306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.29 - 1.82i)T \)
11 \( 1 \)
good2 \( 1 - 1.23iT - 2T^{2} \)
3 \( 1 - 0.363iT - 3T^{2} \)
7 \( 1 - 2.58iT - 7T^{2} \)
13 \( 1 + 2.75iT - 13T^{2} \)
17 \( 1 + 3.85iT - 17T^{2} \)
19 \( 1 - 0.277T + 19T^{2} \)
23 \( 1 + 8.40iT - 23T^{2} \)
29 \( 1 + 3.32T + 29T^{2} \)
31 \( 1 - 0.564T + 31T^{2} \)
37 \( 1 - 0.522iT - 37T^{2} \)
41 \( 1 + 5.11T + 41T^{2} \)
43 \( 1 - 2.54iT - 43T^{2} \)
47 \( 1 - 4.92iT - 47T^{2} \)
53 \( 1 + 8.72iT - 53T^{2} \)
59 \( 1 + 7.50T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 + 3.20iT - 67T^{2} \)
71 \( 1 + 8.40T + 71T^{2} \)
73 \( 1 + 13.0iT - 73T^{2} \)
79 \( 1 + 9.70T + 79T^{2} \)
83 \( 1 - 3.29iT - 83T^{2} \)
89 \( 1 + 2.48T + 89T^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.81628581188652995255924211377, −10.09151023095482852601753571085, −9.162809901694595174913446861061, −8.155714797531000335615096882929, −7.20618651718318998507051364164, −6.51788121943812012223895944352, −5.68235575197049061236699311595, −4.81054446442392977909919129989, −3.05124605669666902601914186215, −2.11752355547260130492552762217, 1.22921144103181858156412248283, 1.91381353749165689576089090637, 3.67242698305647653730054350325, 4.37873508629654223912210624631, 5.77810554144044185715567980923, 6.89359321070953447080622269420, 7.53390828936231474825457478831, 8.856782796531237698143280228584, 9.864060598565904751559886639140, 10.20303708239544257011162051348

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