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Properties

Label 2-418-11.5-c1-0-11
Degree $2$
Conductor $418$
Sign $0.569 - 0.821i$
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 + (1.61 + 1.17i)3-s + (−0.809 + 0.587i)4-s + (0.5 − 1.53i)5-s + (−0.618 + 1.90i)6-s + (3.92 − 2.85i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 1.61·10-s + (−0.809 + 3.21i)11-s − 1.99·12-s + (0.763 + 2.35i)13-s + (3.92 + 2.85i)14-s + (2.61 − 1.90i)15-s + (0.309 − 0.951i)16-s + (1.04 − 3.21i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.934 + 0.678i)3-s + (−0.404 + 0.293i)4-s + (0.223 − 0.688i)5-s + (−0.252 + 0.776i)6-s + (1.48 − 1.07i)7-s + (−0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + 0.511·10-s + (−0.243 + 0.969i)11-s − 0.577·12-s + (0.211 + 0.652i)13-s + (1.04 + 0.762i)14-s + (0.675 − 0.491i)15-s + (0.0772 − 0.237i)16-s + (0.253 − 0.780i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.569 - 0.821i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.569 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97641 + 1.03455i\)
\(L(\frac12)\) \(\approx\) \(1.97641 + 1.03455i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 - 3.21i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-1.61 - 1.17i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.5 + 1.53i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-3.92 + 2.85i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.763 - 2.35i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.04 + 3.21i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + 7.85T + 23T^{2} \)
29 \( 1 + (8.47 - 6.15i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.618 - 1.90i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.23 - 2.35i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.85 - 2.80i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.09T + 43T^{2} \)
47 \( 1 + (1.11 + 0.812i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.47 + 4.53i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.85 + 2.07i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.95 - 9.09i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.76T + 67T^{2} \)
71 \( 1 + (-1.52 + 4.70i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-11.3 + 8.22i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.14 - 3.52i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.26 - 13.1i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + (2.85 + 8.78i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.30838981935419160146676968256, −10.15221709446304125201640623812, −9.379243878701528732883628623813, −8.506918219679847992967217364815, −7.79033381213849726021429186433, −6.90799334665341064683220874788, −5.18921194857340329270528575339, −4.55977361686521505240658822173, −3.71011300161005432616438855563, −1.76828937003339794255804046403, 1.82940283674798730342952289288, 2.51051114997803138718570421131, 3.72469303087348185420765368918, 5.35201652597748484991408689131, 6.12086743482653085557087210220, 7.963888366366607151651566328179, 8.093781975588859108324934516506, 9.089695198523938625530104405380, 10.40521783690185483826907296948, 11.11667657109272319502761295426

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