↑

Properties

Label 2-476-119.10-c1-0-0
Degree $2$
Conductor $476$
Sign $0.734 - 0.678i$
Analytic cond. $3.80087$
Root an. cond. $1.94958$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.29 − 2.61i)3-s + (−2.77 + 2.43i)5-s + (−2.19 − 1.48i)7-s + (−3.35 + 4.36i)9-s + (3.07 − 0.201i)11-s + (2.89 + 2.89i)13-s + (9.93 + 4.11i)15-s + (1.63 + 3.78i)17-s + (−0.417 − 3.17i)19-s + (−1.05 + 7.64i)21-s + (3.87 + 1.91i)23-s + (1.12 − 8.51i)25-s + (7.17 + 1.42i)27-s + (−0.411 − 2.06i)29-s + (−9.89 + 4.87i)31-s + ⋯
L(s)  = 1  + (−0.744 − 1.51i)3-s + (−1.23 + 1.08i)5-s + (−0.828 − 0.560i)7-s + (−1.11 + 1.45i)9-s + (0.928 − 0.0608i)11-s + (0.802 + 0.802i)13-s + (2.56 + 1.06i)15-s + (0.395 + 0.918i)17-s + (−0.0958 − 0.728i)19-s + (−0.230 + 1.66i)21-s + (0.808 + 0.398i)23-s + (0.224 − 1.70i)25-s + (1.38 + 0.274i)27-s + (−0.0764 − 0.384i)29-s + (−1.77 + 0.876i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.734 - 0.678i$
Analytic conductor: \(3.80087\)
Root analytic conductor: \(1.94958\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1/2),\ 0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.525104 + 0.205252i\)
\(L(\frac12)\) \(\approx\) \(0.525104 + 0.205252i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (2.19 + 1.48i)T \)
17 \( 1 + (-1.63 - 3.78i)T \)
good3 \( 1 + (1.29 + 2.61i)T + (-1.82 + 2.38i)T^{2} \)
5 \( 1 + (2.77 - 2.43i)T + (0.652 - 4.95i)T^{2} \)
11 \( 1 + (-3.07 + 0.201i)T + (10.9 - 1.43i)T^{2} \)
13 \( 1 + (-2.89 - 2.89i)T + 13iT^{2} \)
19 \( 1 + (0.417 + 3.17i)T + (-18.3 + 4.91i)T^{2} \)
23 \( 1 + (-3.87 - 1.91i)T + (14.0 + 18.2i)T^{2} \)
29 \( 1 + (0.411 + 2.06i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (9.89 - 4.87i)T + (18.8 - 24.5i)T^{2} \)
37 \( 1 + (0.356 - 5.44i)T + (-36.6 - 4.82i)T^{2} \)
41 \( 1 + (0.189 - 0.950i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (-2.99 - 7.23i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (2.06 - 7.70i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.82 - 10.1i)T + (-13.7 + 51.1i)T^{2} \)
59 \( 1 + (-2.69 - 0.354i)T + (56.9 + 15.2i)T^{2} \)
61 \( 1 + (2.27 + 6.69i)T + (-48.3 + 37.1i)T^{2} \)
67 \( 1 + (3.43 + 1.98i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.51 + 2.34i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (-2.50 - 0.848i)T + (57.9 + 44.4i)T^{2} \)
79 \( 1 + (4.03 - 8.18i)T + (-48.0 - 62.6i)T^{2} \)
83 \( 1 + (-1.17 + 2.84i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-6.34 - 1.70i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (17.5 - 3.49i)T + (89.6 - 37.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.13506326741464138907711515474, −10.81137357950082051544177617780, −9.217494735603565777520215385819, −8.009817682550796980369878892360, −7.12014859001361386781460711225, −6.74333267220915214893348060118, −6.00988537538612382846138396035, −4.11680424411480926318976217924, −3.13448323736637924528873376462, −1.28404982061879582674687459895, 0.43423417239868582539135559144, 3.54257557087002294550942672255, 3.95074072857907089254268145588, 5.19319576150501848419005484503, 5.76977198234102291235037152762, 7.22108150807747788551572196053, 8.676090069685210396982432026270, 9.054264725295180727035136206935, 9.981982648258715597899947226337, 10.97847426503287019641692620513

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