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Properties

Label 2-418-11.4-c1-0-2
Degree $2$
Conductor $418$
Sign $-0.986 + 0.164i$
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  + (0.809 + 0.587i)2-s + (−0.784 + 2.41i)3-s + (0.309 + 0.951i)4-s + (−1.46 + 1.06i)5-s + (−2.05 + 1.49i)6-s + (−0.0865 − 0.266i)7-s + (−0.309 + 0.951i)8-s + (−2.79 − 2.02i)9-s − 1.81·10-s + (−3.28 + 0.466i)11-s − 2.53·12-s + (0.680 + 0.494i)13-s + (0.0865 − 0.266i)14-s + (−1.42 − 4.38i)15-s + (−0.809 + 0.587i)16-s + (3.48 − 2.53i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.453 + 1.39i)3-s + (0.154 + 0.475i)4-s + (−0.656 + 0.477i)5-s + (−0.838 + 0.609i)6-s + (−0.0327 − 0.100i)7-s + (−0.109 + 0.336i)8-s + (−0.930 − 0.675i)9-s − 0.573·10-s + (−0.990 + 0.140i)11-s − 0.733·12-s + (0.188 + 0.137i)13-s + (0.0231 − 0.0712i)14-s + (−0.367 − 1.13i)15-s + (−0.202 + 0.146i)16-s + (0.844 − 0.613i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.986 + 0.164i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.986 + 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0927198 - 1.12171i\)
\(L(\frac12)\) \(\approx\) \(0.0927198 - 1.12171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (3.28 - 0.466i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.784 - 2.41i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.46 - 1.06i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.0865 + 0.266i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.680 - 0.494i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.48 + 2.53i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 - 3.84T + 23T^{2} \)
29 \( 1 + (-2.44 - 7.52i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (7.24 + 5.26i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.494 + 1.52i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.18 - 6.73i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.63T + 43T^{2} \)
47 \( 1 + (2.36 - 7.27i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.34 - 5.33i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.14 - 3.52i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.143 + 0.104i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 8.01T + 67T^{2} \)
71 \( 1 + (3.04 - 2.21i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.184 - 0.567i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-11.4 - 8.33i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.61 - 1.90i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (7.80 + 5.67i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.42820000365986249269466539632, −10.88752619201353715985699762818, −10.05580974982080103562291958556, −9.064150543593959116241228338828, −7.82173011945164461619894573307, −7.01229487518103419161471741942, −5.59315558722787413450271960888, −4.97417372910881724732477657510, −3.88285000219127818996444132940, −3.05303144749993745574573074347, 0.63366864167628086617366911409, 2.12066709588287853531724615383, 3.57014573820010165969123945610, 5.03766834047235690221026158325, 5.86295162526101894593546747203, 6.94334330146537264688653566834, 7.82974646198687241330492164774, 8.611388632296286062055551957945, 10.15136717125148220744748402938, 11.04453756237777470828358844573

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