Properties

Label 2-418-209.201-c1-0-19
Degree $2$
Conductor $418$
Sign $-0.397 - 0.917i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
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  + (−0.104 − 0.994i)2-s + (−1.95 − 0.415i)3-s + (−0.978 + 0.207i)4-s + (−0.348 + 0.155i)5-s + (−0.209 + 1.98i)6-s + (1.30 − 4.02i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (0.190 + 0.330i)10-s + (−2.54 + 2.12i)11-s + 2·12-s + (−1.69 − 0.754i)13-s + (−4.14 − 0.880i)14-s + (0.747 − 0.158i)15-s + (0.913 − 0.406i)16-s + (−4.91 + 2.18i)17-s + ⋯
L(s)  = 1  + (−0.0739 − 0.703i)2-s + (−1.12 − 0.240i)3-s + (−0.489 + 0.103i)4-s + (−0.156 + 0.0694i)5-s + (−0.0853 + 0.812i)6-s + (0.494 − 1.52i)7-s + (0.109 + 0.336i)8-s + (0.304 + 0.135i)9-s + (0.0603 + 0.104i)10-s + (−0.767 + 0.641i)11-s + 0.577·12-s + (−0.469 − 0.209i)13-s + (−1.10 − 0.235i)14-s + (0.192 − 0.0410i)15-s + (0.228 − 0.101i)16-s + (−1.19 + 0.530i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.397 - 0.917i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.397 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0442643 + 0.0673902i\)
\(L(\frac12)\) \(\approx\) \(0.0442643 + 0.0673902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.104 + 0.994i)T \)
11 \( 1 + (2.54 - 2.12i)T \)
19 \( 1 + (-4.30 - 0.705i)T \)
good3 \( 1 + (1.95 + 0.415i)T + (2.74 + 1.22i)T^{2} \)
5 \( 1 + (0.348 - 0.155i)T + (3.34 - 3.71i)T^{2} \)
7 \( 1 + (-1.30 + 4.02i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.69 + 0.754i)T + (8.69 + 9.66i)T^{2} \)
17 \( 1 + (4.91 - 2.18i)T + (11.3 - 12.6i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.18 - 0.464i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (7.97 - 5.79i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.04 - 3.21i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (9.72 + 2.06i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (-5.35 + 9.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.669 + 0.743i)T + (-4.91 - 46.7i)T^{2} \)
53 \( 1 + (-6.17 - 2.75i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (-5.82 - 6.47i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (-0.746 + 7.10i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (6.16 + 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.77 - 2.57i)T + (47.5 - 52.7i)T^{2} \)
73 \( 1 + (7.51 + 8.35i)T + (-7.63 + 72.6i)T^{2} \)
79 \( 1 + (0.387 + 3.68i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (9.28 + 6.74i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (7.35 + 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.69 - 16.0i)T + (-94.8 + 20.1i)T^{2} \)
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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

−10.60822988167301083686725755076, −10.25612731066801623066465071992, −8.894547743997840051566627988311, −7.50171902265748854260713114402, −7.07410397662223478824055258495, −5.50595141392848640874600943952, −4.71538703535730862670893603396, −3.55694722992266071781308404947, −1.65890746341467597358577895229, −0.05992579775921390997727502779, 2.52344786582075759018785215516, 4.50263478335405823907397344553, 5.48230144855062192851128943343, 5.77443563896649471153199710129, 7.06003328375395667225509283170, 8.199134863974203194360896742435, 8.970742657046593549233336290934, 9.945292671833500830634810767119, 11.30434687811209771005793544415, 11.49913479339809566032170658829

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