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Properties

Label 2-418-11.9-c1-0-15
Degree $2$
Conductor $418$
Sign $-0.957 + 0.288i$
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  + (0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.809 − 2.48i)5-s + (−0.309 − 0.951i)6-s + (−1.30 − 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.618 + 1.90i)9-s − 2.61·10-s + (−3.30 − 0.224i)11-s − 0.999·12-s + (1.5 − 4.61i)13-s + (−1.30 + 0.951i)14-s + (−2.11 − 1.53i)15-s + (0.309 + 0.951i)16-s + (0.0729 + 0.224i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.361 − 1.11i)5-s + (−0.126 − 0.388i)6-s + (−0.494 − 0.359i)7-s + (−0.286 + 0.207i)8-s + (−0.206 + 0.634i)9-s − 0.827·10-s + (−0.997 − 0.0676i)11-s − 0.288·12-s + (0.416 − 1.28i)13-s + (−0.349 + 0.254i)14-s + (−0.546 − 0.397i)15-s + (0.0772 + 0.237i)16-s + (0.0176 + 0.0544i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.957 + 0.288i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.957 + 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180073 - 1.22386i\)
\(L(\frac12)\) \(\approx\) \(0.180073 - 1.22386i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (3.30 + 0.224i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-0.809 + 0.587i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.809 + 2.48i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (1.30 + 0.951i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.5 + 4.61i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.0729 - 0.224i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + (-7.23 - 5.25i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.69 + 8.28i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (7.59 + 5.51i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.92 + 3.57i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 7.70T + 43T^{2} \)
47 \( 1 + (-7.04 + 5.11i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.39 + 4.30i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.73 - 3.44i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.5 - 13.8i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 5.61T + 67T^{2} \)
71 \( 1 + (-3.01 - 9.28i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.309 + 0.224i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.63 + 14.2i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.71 - 8.36i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 18.4T + 89T^{2} \)
97 \( 1 + (-2.83 + 8.73i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.63964981686066060863409082447, −10.16534572096260969802565924312, −8.758699138188202179933120608732, −8.287527097581350579973117213278, −7.33482016811807422207845765372, −5.64632014644964590476436907646, −4.89672868521433413411625018028, −3.58225279426673764975951211412, −2.44866174255394761011154969386, −0.70317471493933793905103311929, 2.76460849092801612630537335210, 3.55573412870410379632275685279, 4.78973988911871631448435624754, 6.29525711451334411555332160402, 6.72808369717639118892167725574, 7.921938183946382418602047827117, 8.784671602988928896611541492263, 9.714237894926524668422130898514, 10.57165772302175844236805683037, 11.70860574640636347925659990699

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