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Properties

Label 2-418-11.4-c1-0-8
Degree $2$
Conductor $418$
Sign $-0.0694 + 0.997i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (−1 + 3.07i)3-s + (0.309 + 0.951i)4-s + (−3.11 + 2.26i)5-s + (2.61 − 1.90i)6-s + (−1.19 − 3.66i)7-s + (0.309 − 0.951i)8-s + (−6.04 − 4.39i)9-s + 3.85·10-s + (−0.309 + 3.30i)11-s − 3.23·12-s + (3.23 + 2.35i)13-s + (−1.19 + 3.66i)14-s + (−3.85 − 11.8i)15-s + (−0.809 + 0.587i)16-s + (−1.30 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.577 + 1.77i)3-s + (0.154 + 0.475i)4-s + (−1.39 + 1.01i)5-s + (1.06 − 0.776i)6-s + (−0.450 − 1.38i)7-s + (0.109 − 0.336i)8-s + (−2.01 − 1.46i)9-s + 1.21·10-s + (−0.0931 + 0.995i)11-s − 0.934·12-s + (0.897 + 0.652i)13-s + (−0.318 + 0.979i)14-s + (−0.995 − 3.06i)15-s + (−0.202 + 0.146i)16-s + (−0.317 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0694 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.0694 + 0.997i$
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: \(1\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.0694 + 0.997i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 3.30i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (1 - 3.07i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (3.11 - 2.26i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.19 + 3.66i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.23 - 2.35i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.30 - 0.951i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 - 0.854T + 23T^{2} \)
29 \( 1 + (1.76 + 5.42i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.23 - 1.62i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1 + 3.07i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.145 + 0.449i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.61T + 43T^{2} \)
47 \( 1 + (0.263 - 0.812i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.09 + 6.60i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.14 - 6.60i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.927 + 0.673i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.23T + 67T^{2} \)
71 \( 1 + (-2.23 + 1.62i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.38 + 10.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.47 - 5.42i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.73 - 1.26i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + (13.7 + 9.95i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.94319270031492454655469383823, −10.21814967995834433649101753391, −9.592921257079716724719451461039, −8.393504742109723317316828193058, −7.24616582790049482804533822521, −6.45280128849526491843161783232, −4.49701347906267106458992470949, −3.97877204515060317220210693074, −3.23843910963168283916880220277, 0, 1.26529511227840284060139585803, 3.05790286250575764576336802142, 5.21873118987883576079609004089, 5.91922949582519498270890785764, 6.83440790618441897955124767877, 8.011123253548584954673435879010, 8.347840917476763219714302354061, 9.028830481409135347769299651085, 10.99695671372290293112477525412

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