Properties

Label 2-418-11.9-c1-0-2
Degree $2$
Conductor $418$
Sign $-0.998 - 0.0626i$
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 + (−1.69 + 1.23i)3-s + (−0.809 − 0.587i)4-s + (0.760 + 2.33i)5-s + (−0.649 − 1.99i)6-s + (2.50 + 1.82i)7-s + (0.809 − 0.587i)8-s + (0.436 − 1.34i)9-s − 2.46·10-s + (3.02 + 1.36i)11-s + 2.10·12-s + (−1.22 + 3.78i)13-s + (−2.50 + 1.82i)14-s + (−4.18 − 3.03i)15-s + (0.309 + 0.951i)16-s + (−0.663 − 2.04i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.981 + 0.712i)3-s + (−0.404 − 0.293i)4-s + (0.339 + 1.04i)5-s + (−0.265 − 0.815i)6-s + (0.947 + 0.688i)7-s + (0.286 − 0.207i)8-s + (0.145 − 0.448i)9-s − 0.777·10-s + (0.912 + 0.410i)11-s + 0.606·12-s + (−0.340 + 1.04i)13-s + (−0.670 + 0.486i)14-s + (−1.07 − 0.784i)15-s + (0.0772 + 0.237i)16-s + (−0.160 − 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.998 - 0.0626i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.998 - 0.0626i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0290120 + 0.925185i\)
\(L(\frac12)\) \(\approx\) \(0.0290120 + 0.925185i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.02 - 1.36i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (1.69 - 1.23i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.760 - 2.33i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.50 - 1.82i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.22 - 3.78i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.663 + 2.04i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + 2.55T + 23T^{2} \)
29 \( 1 + (5.26 + 3.82i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.459 + 1.41i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.41 - 1.02i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.43 - 3.22i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.59T + 43T^{2} \)
47 \( 1 + (-2.75 + 2.00i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.63 - 11.1i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.38 - 3.90i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.95 + 6.03i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + (4.93 + 15.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.56 - 1.86i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.24 + 6.90i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.46 - 10.6i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 4.63T + 89T^{2} \)
97 \( 1 + (1.69 - 5.20i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.50008216755262564953808323530, −10.79423725514712879064865090472, −9.768989974794140247863834556908, −9.153123761281986666837075325081, −7.81876393192418127684558789936, −6.76497513640786096970439058823, −6.03052516242555537325177617903, −5.03838433061437983196216226021, −4.17032405935729145759211406819, −2.13634850328721266566042569101, 0.78603850719315197454633668167, 1.65184703520360715659472219547, 3.78974505599213232350647683528, 5.02243355524755163103217775080, 5.76954654999295853691811762803, 7.07703730334696660518952001147, 8.100488341660568863600864233336, 8.935701051028451626125231853300, 10.04150499957512801260498234381, 11.03914431596421133156192954845

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