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Properties

Label 2-418-11.4-c1-0-16
Degree $2$
Conductor $418$
Sign $-0.908 + 0.417i$
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 + (0.690 − 2.12i)3-s + (0.309 + 0.951i)4-s + (1.30 − 0.951i)5-s + (−1.80 + 1.31i)6-s + (−1.19 − 3.66i)7-s + (0.309 − 0.951i)8-s + (−1.61 − 1.17i)9-s − 1.61·10-s + (−3.04 + 1.31i)11-s + 2.23·12-s + (0.5 + 0.363i)13-s + (−1.19 + 3.66i)14-s + (−1.11 − 3.44i)15-s + (−0.809 + 0.587i)16-s + (1.42 − 1.03i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.398 − 1.22i)3-s + (0.154 + 0.475i)4-s + (0.585 − 0.425i)5-s + (−0.738 + 0.536i)6-s + (−0.450 − 1.38i)7-s + (0.109 − 0.336i)8-s + (−0.539 − 0.391i)9-s − 0.511·10-s + (−0.918 + 0.396i)11-s + 0.645·12-s + (0.138 + 0.100i)13-s + (−0.318 + 0.979i)14-s + (−0.288 − 0.888i)15-s + (−0.202 + 0.146i)16-s + (0.346 − 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.908 + 0.417i$
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: \((2,\ 418,\ (\ :1/2),\ -0.908 + 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.243185 - 1.11030i\)
\(L(\frac12)\) \(\approx\) \(0.243185 - 1.11030i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.04 - 1.31i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (-0.690 + 2.12i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.30 + 0.951i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.19 + 3.66i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.5 - 0.363i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.42 + 1.03i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + (0.472 + 1.45i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.19 + 1.59i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.354 + 1.08i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.19 - 6.74i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.70T + 43T^{2} \)
47 \( 1 + (-0.781 + 2.40i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-9.66 - 7.02i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.590 + 1.81i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.35 + 3.88i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 5.09T + 67T^{2} \)
71 \( 1 + (-6.66 + 4.84i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.42 + 13.6i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.42 + 1.76i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.16 + 5.93i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + (-8.42 - 6.12i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.63213442274900609524661853266, −9.946747639053860453595040333860, −9.013723071943378529427288330985, −7.85848563299355928948342703724, −7.33963685664800012518430944479, −6.49298818026468966313307820695, −4.94688603928653644385442678822, −3.37062197010359174089942532270, −2.04663983274109719744727023191, −0.854791012166043315870259358163, 2.40865985519220425276816131503, 3.41891448035543341709797459629, 5.15054567338486562520086448513, 5.74671846119690708713383112615, 6.89252941607184658502267321408, 8.386690577171578155865124820456, 8.863444109406005128690707991754, 9.852469240485717282773187782248, 10.25811975071947240501171026237, 11.20892372071604621922391867918

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