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Properties

Label 2-418-209.65-c1-0-19
Degree $2$
Conductor $418$
Sign $-0.730 - 0.682i$
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  + (0.5 − 0.866i)2-s + (−1.75 − 1.01i)3-s + (−0.499 − 0.866i)4-s + (−0.202 + 0.350i)5-s + (−1.75 + 1.01i)6-s − 1.38i·7-s − 0.999·8-s + (0.550 + 0.953i)9-s + (0.202 + 0.350i)10-s + (−2.65 + 1.99i)11-s + 2.02i·12-s + (−1.16 − 2.02i)13-s + (−1.19 − 0.692i)14-s + (0.709 − 0.409i)15-s + (−0.5 + 0.866i)16-s + (−4.98 − 2.87i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−1.01 − 0.584i)3-s + (−0.249 − 0.433i)4-s + (−0.0904 + 0.156i)5-s + (−0.716 + 0.413i)6-s − 0.523i·7-s − 0.353·8-s + (0.183 + 0.317i)9-s + (0.0639 + 0.110i)10-s + (−0.799 + 0.600i)11-s + 0.584i·12-s + (−0.323 − 0.561i)13-s + (−0.320 − 0.185i)14-s + (0.183 − 0.105i)15-s + (−0.125 + 0.216i)16-s + (−1.20 − 0.697i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 - 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.730 - 0.682i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.730 - 0.682i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129018 + 0.326904i\)
\(L(\frac12)\) \(\approx\) \(0.129018 + 0.326904i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (2.65 - 1.99i)T \)
19 \( 1 + (1.88 - 3.93i)T \)
good3 \( 1 + (1.75 + 1.01i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.202 - 0.350i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.38iT - 7T^{2} \)
13 \( 1 + (1.16 + 2.02i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.98 + 2.87i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.24 - 3.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.87 + 6.70i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.93iT - 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 + (-2.30 + 3.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.81 + 5.66i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.27 + 9.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.60 - 2.08i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.94 + 1.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.11 + 1.21i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.42 + 4.28i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.32 + 3.07i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.74 + 4.47i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.32 + 7.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.841iT - 83T^{2} \)
89 \( 1 + (12.4 - 7.18i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.2 - 7.62i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.84674031771749712486879505296, −10.14432703572480574154024241724, −8.994961888419399797500191534398, −7.52813253265581207504755268724, −6.87570088538039577603903683153, −5.65217836454221214171705181022, −4.91149672428926232279045150294, −3.53604165094030741404849401469, −1.98274105434092316156499233914, −0.21505909697675360981356128215, 2.69968177792647512685839726205, 4.47118220099639342653284394900, 4.94406105075070230977878937420, 6.06799074271878898459420120401, 6.71841817364006709374872270817, 8.189420597957905557052983057892, 8.867824297027036988064652115660, 10.07517823281757315325765325816, 11.09689965000515756589387951813, 11.52395851614766131658556997539

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