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Properties

Label 2-418-11.3-c1-0-16
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: $\chi_{418} (267, \cdot )$
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

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

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