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Properties

Label 2-476-119.108-c1-0-2
Degree $2$
Conductor $476$
Sign $0.680 - 0.732i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.81 − 0.617i)3-s + (−0.0377 + 0.576i)5-s + (−2.49 − 0.892i)7-s + (0.546 + 0.419i)9-s + (2.79 + 2.45i)11-s + (1.80 + 1.80i)13-s + (0.424 − 1.02i)15-s + (4.02 + 0.873i)17-s + (−4.67 + 0.614i)19-s + (3.97 + 3.16i)21-s + (1.83 + 5.41i)23-s + (4.62 + 0.609i)25-s + (2.46 + 3.69i)27-s + (0.603 + 0.402i)29-s + (1.62 − 4.79i)31-s + ⋯
L(s)  = 1  + (−1.04 − 0.356i)3-s + (−0.0168 + 0.257i)5-s + (−0.941 − 0.337i)7-s + (0.182 + 0.139i)9-s + (0.844 + 0.740i)11-s + (0.501 + 0.501i)13-s + (0.109 − 0.264i)15-s + (0.977 + 0.211i)17-s + (−1.07 + 0.141i)19-s + (0.868 + 0.689i)21-s + (0.383 + 1.12i)23-s + (0.925 + 0.121i)25-s + (0.474 + 0.710i)27-s + (0.111 + 0.0748i)29-s + (0.292 − 0.860i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.680 - 0.732i$
Analytic conductor: \(3.80087\)
Root analytic conductor: \(1.94958\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (465, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1/2),\ 0.680 - 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738222 + 0.322030i\)
\(L(\frac12)\) \(\approx\) \(0.738222 + 0.322030i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.49 + 0.892i)T \)
17 \( 1 + (-4.02 - 0.873i)T \)
good3 \( 1 + (1.81 + 0.617i)T + (2.38 + 1.82i)T^{2} \)
5 \( 1 + (0.0377 - 0.576i)T + (-4.95 - 0.652i)T^{2} \)
11 \( 1 + (-2.79 - 2.45i)T + (1.43 + 10.9i)T^{2} \)
13 \( 1 + (-1.80 - 1.80i)T + 13iT^{2} \)
19 \( 1 + (4.67 - 0.614i)T + (18.3 - 4.91i)T^{2} \)
23 \( 1 + (-1.83 - 5.41i)T + (-18.2 + 14.0i)T^{2} \)
29 \( 1 + (-0.603 - 0.402i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (-1.62 + 4.79i)T + (-24.5 - 18.8i)T^{2} \)
37 \( 1 + (-7.87 - 8.98i)T + (-4.82 + 36.6i)T^{2} \)
41 \( 1 + (7.82 - 5.22i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (4.23 - 1.75i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + (0.714 - 2.66i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-11.3 + 8.72i)T + (13.7 - 51.1i)T^{2} \)
59 \( 1 + (-1.40 + 10.6i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (3.58 - 7.27i)T + (-37.1 - 48.3i)T^{2} \)
67 \( 1 + (-6.52 - 3.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.23 - 0.644i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (11.7 - 5.77i)T + (44.4 - 57.9i)T^{2} \)
79 \( 1 + (6.72 - 2.28i)T + (62.6 - 48.0i)T^{2} \)
83 \( 1 + (-2.41 - 0.999i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (14.4 + 3.87i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.45 + 6.66i)T + (-37.1 - 89.6i)T^{2} \)
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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

−11.36768299219583209624279100925, −10.21443822003466256617121171265, −9.596485411526813113164741920309, −8.409390235361697658060714586978, −7.00452058907404740894664551752, −6.57746192393264196821470953494, −5.73514044050195282206497858431, −4.39588294526803059971961850537, −3.23276491083876288318909258833, −1.28071353141911079302935667211, 0.65191676738468705925793455547, 2.92929985623958690504192029096, 4.16487353196909155072036001837, 5.38886517164432359448362362200, 6.09939447066801511208497031042, 6.85531212224080474430909631621, 8.458746482145250516377157911650, 9.034657024351003590309328465332, 10.33311100879194641972123181139, 10.72203785886661767957197967916

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