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Properties

Label 2-190-5.4-c1-0-6
Degree $2$
Conductor $190$
Sign $0.948 + 0.316i$
Analytic cond. $1.51715$
Root an. cond. $1.23172$
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  + i·2-s − 0.414i·3-s − 4-s + (0.707 − 2.12i)5-s + 0.414·6-s − 4.41i·7-s − i·8-s + 2.82·9-s + (2.12 + 0.707i)10-s − 1.41·11-s + 0.414i·12-s + 5.82i·13-s + 4.41·14-s + (−0.878 − 0.292i)15-s + 16-s − i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.239i·3-s − 0.5·4-s + (0.316 − 0.948i)5-s + 0.169·6-s − 1.66i·7-s − 0.353i·8-s + 0.942·9-s + (0.670 + 0.223i)10-s − 0.426·11-s + 0.119i·12-s + 1.61i·13-s + 1.17·14-s + (−0.226 − 0.0756i)15-s + 0.250·16-s − 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $0.948 + 0.316i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{190} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 190,\ (\ :1/2),\ 0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17725 - 0.191042i\)
\(L(\frac12)\) \(\approx\) \(1.17725 - 0.191042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-0.707 + 2.12i)T \)
19 \( 1 - T \)
good3 \( 1 + 0.414iT - 3T^{2} \)
7 \( 1 + 4.41iT - 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 - 5.82iT - 13T^{2} \)
17 \( 1 + iT - 17T^{2} \)
23 \( 1 - 0.757iT - 23T^{2} \)
29 \( 1 - 0.171T + 29T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 - 8.48iT - 37T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 - 1.75iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 5.48iT - 53T^{2} \)
59 \( 1 + 6.89T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 + 4.75iT - 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 11.4iT - 73T^{2} \)
79 \( 1 - 6.48T + 79T^{2} \)
83 \( 1 - 14.4iT - 83T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 - 0.343iT - 97T^{2} \)
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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

−12.84026262976644013734149674382, −11.66709833529382198081184685133, −10.20811269563823693358341426021, −9.555864716288353031490040441524, −8.273651025419585526355966086043, −7.26595641815642898297319851997, −6.51151305988833745420259071204, −4.80663747676699940447118266967, −4.13756085135327951424793360402, −1.27961502630059768149103804813, 2.28095108453774903385387795096, 3.31919267666252988739955505355, 5.12621272769236787494226985625, 6.07929888080104895390591115765, 7.64908703923777749395982372292, 8.808212735528691310948521915042, 9.968751386454530848925602530910, 10.47766264112560788309307808799, 11.61628367479862979123327885029, 12.58310225874120778139327131706

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