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Properties

Label 2-418-11.3-c1-0-7
Degree $2$
Conductor $418$
Sign $0.642 + 0.766i$
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 + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.224i)5-s + (0.809 + 0.587i)6-s + (−0.190 + 0.587i)7-s + (0.309 + 0.951i)8-s + (1.61 − 1.17i)9-s − 0.381·10-s + (−2.19 − 2.48i)11-s − 12-s + (1.5 − 1.08i)13-s + (−0.190 − 0.587i)14-s + (0.118 − 0.363i)15-s + (−0.809 − 0.587i)16-s + (3.42 + 2.48i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (0.138 + 0.100i)5-s + (0.330 + 0.239i)6-s + (−0.0721 + 0.222i)7-s + (0.109 + 0.336i)8-s + (0.539 − 0.391i)9-s − 0.120·10-s + (−0.660 − 0.750i)11-s − 0.288·12-s + (0.416 − 0.302i)13-s + (−0.0510 − 0.157i)14-s + (0.0304 − 0.0937i)15-s + (−0.202 − 0.146i)16-s + (0.831 + 0.603i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.642 + 0.766i$
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: \((2,\ 418,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.888196 - 0.414324i\)
\(L(\frac12)\) \(\approx\) \(0.888196 - 0.414324i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 2.48i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (0.309 + 0.951i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.309 - 0.224i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.190 - 0.587i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.5 + 1.08i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.42 - 2.48i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + (-2.76 + 8.50i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.80 + 2.76i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.59 + 11.0i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.57 - 4.84i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 + (-1.45 - 4.47i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.8 - 7.91i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.263 + 0.812i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.5 - 3.26i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.38T + 67T^{2} \)
71 \( 1 + (11.5 + 8.36i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.809 + 2.48i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (12.1 - 8.81i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-12.7 - 9.28i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.41T + 89T^{2} \)
97 \( 1 + (-10.6 + 7.74i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.92210189698481551197331454175, −10.12015640713493719511635035555, −9.245538212579627077117997626986, −8.063519277483862446072754962172, −7.60654740430458773254282084608, −6.10861200329363040806316366733, −5.97240282200856395122011021252, −4.20492094617288137901934527553, −2.54934059175460208798083824939, −0.855896443166178480029742274647, 1.56832581894284171788253727077, 3.15460995055558064386582485501, 4.44063736686985612816622519737, 5.37346365615988541958530190702, 6.91238814191365402185769288964, 7.69397523502468456027529624980, 8.773155991996078103552158787849, 9.859347291012809419722591925133, 10.20637734491738217628039662300, 11.10954788497004654096130559715

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