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Properties

Label 2-418-209.189-c1-0-0
Degree $2$
Conductor $418$
Sign $-0.997 - 0.0686i$
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  + (−0.309 + 0.951i)2-s + (0.943 + 1.29i)3-s + (−0.809 − 0.587i)4-s + (−0.757 − 2.32i)5-s + (−1.52 + 0.495i)6-s + (−2.75 + 3.79i)7-s + (0.809 − 0.587i)8-s + (0.131 − 0.403i)9-s + 2.44·10-s + (−1.86 + 2.74i)11-s − 1.60i·12-s + (−0.993 + 3.05i)13-s + (−2.75 − 3.79i)14-s + (2.31 − 3.18i)15-s + (0.309 + 0.951i)16-s + (−6.01 + 1.95i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.544 + 0.749i)3-s + (−0.404 − 0.293i)4-s + (−0.338 − 1.04i)5-s + (−0.623 + 0.202i)6-s + (−1.04 + 1.43i)7-s + (0.286 − 0.207i)8-s + (0.0436 − 0.134i)9-s + 0.774·10-s + (−0.562 + 0.826i)11-s − 0.463i·12-s + (−0.275 + 0.848i)13-s + (−0.736 − 1.01i)14-s + (0.596 − 0.821i)15-s + (0.0772 + 0.237i)16-s + (−1.45 + 0.474i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.997 - 0.0686i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.997 - 0.0686i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0260441 + 0.758142i\)
\(L(\frac12)\) \(\approx\) \(0.0260441 + 0.758142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (1.86 - 2.74i)T \)
19 \( 1 + (1.66 - 4.02i)T \)
good3 \( 1 + (-0.943 - 1.29i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.757 + 2.32i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.75 - 3.79i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.993 - 3.05i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (6.01 - 1.95i)T + (13.7 - 9.99i)T^{2} \)
23 \( 1 + 0.141T + 23T^{2} \)
29 \( 1 + (1.90 + 1.38i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-7.44 - 2.41i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.80 - 3.86i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.41 + 5.38i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6.62iT - 43T^{2} \)
47 \( 1 + (4.37 - 3.18i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-9.91 - 3.22i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.81 + 9.38i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.30 - 1.72i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 14.8iT - 67T^{2} \)
71 \( 1 + (13.3 - 4.33i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.815 + 1.12i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.69 - 8.29i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.80 + 1.23i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 3.81iT - 89T^{2} \)
97 \( 1 + (-0.0710 - 0.0230i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.93307832291558429573190381528, −10.24857592227469271425930815734, −9.563822017714839192031967481138, −8.760978174465359603387422930111, −8.509537513837579554083854265488, −6.93546379663872515906062665074, −5.98217440227597552699408975845, −4.79823392645045784264976370649, −4.01388089816354171024231446355, −2.36322212408901684107907571442, 0.46920550218655704525011953468, 2.61241960264385681641199759159, 3.19465078570713541825372250665, 4.50883046927645964894808673652, 6.43282917609360744657109689830, 7.20919901854806368655063613334, 7.82864350629771087060213644899, 8.952676303101112690701808638296, 10.18895383268079138461288172933, 10.71704719980958784664024832398

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