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Properties

Label 2-418-209.189-c1-0-12
Degree $2$
Conductor $418$
Sign $0.898 - 0.438i$
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 + (1.24 + 1.71i)3-s + (−0.809 − 0.587i)4-s + (−0.810 − 2.49i)5-s + (−2.01 + 0.654i)6-s + (2.23 − 3.07i)7-s + (0.809 − 0.587i)8-s + (−0.459 + 1.41i)9-s + 2.62·10-s + (3.29 − 0.385i)11-s − 2.11i·12-s + (0.226 − 0.698i)13-s + (2.23 + 3.07i)14-s + (3.26 − 4.49i)15-s + (0.309 + 0.951i)16-s + (−2.93 + 0.954i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.718 + 0.989i)3-s + (−0.404 − 0.293i)4-s + (−0.362 − 1.11i)5-s + (−0.822 + 0.267i)6-s + (0.843 − 1.16i)7-s + (0.286 − 0.207i)8-s + (−0.153 + 0.471i)9-s + 0.829·10-s + (0.993 − 0.116i)11-s − 0.611i·12-s + (0.0629 − 0.193i)13-s + (0.596 + 0.821i)14-s + (0.843 − 1.16i)15-s + (0.0772 + 0.237i)16-s + (−0.712 + 0.231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.898 - 0.438i$
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: \((2,\ 418,\ (\ :1/2),\ 0.898 - 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53457 + 0.354500i\)
\(L(\frac12)\) \(\approx\) \(1.53457 + 0.354500i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.29 + 0.385i)T \)
19 \( 1 + (-3.63 - 2.39i)T \)
good3 \( 1 + (-1.24 - 1.71i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.810 + 2.49i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.23 + 3.07i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.226 + 0.698i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.93 - 0.954i)T + (13.7 - 9.99i)T^{2} \)
23 \( 1 + 1.91T + 23T^{2} \)
29 \( 1 + (4.18 + 3.04i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.73 + 0.562i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.05 - 4.21i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.29 - 5.30i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 7.16iT - 43T^{2} \)
47 \( 1 + (-1.67 + 1.21i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-10.8 - 3.52i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.48 - 2.03i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-12.2 + 3.99i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 11.2iT - 67T^{2} \)
71 \( 1 + (-5.05 + 1.64i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.28 - 1.76i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.63 - 8.10i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (12.5 - 4.09i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + (-11.6 - 3.79i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.10177686367186002366666655124, −10.03077730878982880111412907101, −9.339052264375997747004770218865, −8.456241877011405731713057118782, −7.941591900022314763583993345416, −6.73497312642637014119318816830, −5.22481449255571593171246160678, −4.29609405727672182402860459514, −3.79209455381769977475065024498, −1.21585280401568431016456386435, 1.77662170423319010434144150922, 2.57272514298445553688426734913, 3.74918535255342999271708531195, 5.34602612823657075071363335455, 6.89420816263188316776321329641, 7.37801917387552156049771519196, 8.647202200056678176903244696012, 8.974106315804778485433881906565, 10.37650678785553515018120450846, 11.50867833624390545395295543224

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