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Properties

Label 2-418-19.11-c1-0-12
Degree $2$
Conductor $418$
Sign $-0.658 + 0.752i$
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  + (0.5 − 0.866i)2-s + (1 − 1.73i)3-s + (−0.499 − 0.866i)4-s + (1.5 − 2.59i)5-s + (−0.999 − 1.73i)6-s − 7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.5 − 2.59i)10-s + 11-s − 1.99·12-s + (2 + 3.46i)13-s + (−0.5 + 0.866i)14-s + (−3 − 5.19i)15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 − 0.433i)4-s + (0.670 − 1.16i)5-s + (−0.408 − 0.707i)6-s − 0.377·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.474 − 0.821i)10-s + 0.301·11-s − 0.577·12-s + (0.554 + 0.960i)13-s + (−0.133 + 0.231i)14-s + (−0.774 − 1.34i)15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.855300 - 1.88627i\)
\(L(\frac12)\) \(\approx\) \(0.855300 - 1.88627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 - T \)
19 \( 1 + (3.5 - 2.59i)T \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + T + 7T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.5 + 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.98353865382120255724170948324, −9.877362423057999865143123180099, −8.924686139704274987518687740663, −8.467031651044092260009441276899, −7.03620393887706903165464499772, −6.11317790269323866411209143952, −4.95985926861868640547139133272, −3.75231918841972684403261549152, −2.15524718829239426754884768100, −1.33074000930363262732815613750, 2.72105439189372660242162823600, 3.51764230267867490351764021950, 4.64185985594449241551876473203, 6.03595373463731358190879665573, 6.59567123484551413667535859853, 7.84382001674572726484910320123, 8.904507211767582950778476744323, 9.736260757540450817237931022684, 10.43676169019491811916616801470, 11.25664942521452835462549044770

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