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Properties

Label 2-418-209.65-c1-0-2
Degree $2$
Conductor $418$
Sign $0.436 - 0.899i$
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 + (0.347 + 0.200i)3-s + (−0.499 − 0.866i)4-s + (−1.32 + 2.30i)5-s + (0.347 − 0.200i)6-s + 3.90i·7-s − 0.999·8-s + (−1.41 − 2.45i)9-s + (1.32 + 2.30i)10-s + (−1.46 + 2.97i)11-s − 0.401i·12-s + (1.71 + 2.96i)13-s + (3.37 + 1.95i)14-s + (−0.923 + 0.533i)15-s + (−0.5 + 0.866i)16-s + (1.44 + 0.835i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.200 + 0.115i)3-s + (−0.249 − 0.433i)4-s + (−0.594 + 1.02i)5-s + (0.141 − 0.0819i)6-s + 1.47i·7-s − 0.353·8-s + (−0.473 − 0.819i)9-s + (0.420 + 0.728i)10-s + (−0.440 + 0.897i)11-s − 0.115i·12-s + (0.474 + 0.822i)13-s + (0.903 + 0.521i)14-s + (−0.238 + 0.137i)15-s + (−0.125 + 0.216i)16-s + (0.351 + 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.436 - 0.899i$
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: \((2,\ 418,\ (\ :1/2),\ 0.436 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08183 + 0.677381i\)
\(L(\frac12)\) \(\approx\) \(1.08183 + 0.677381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.46 - 2.97i)T \)
19 \( 1 + (3.55 + 2.52i)T \)
good3 \( 1 + (-0.347 - 0.200i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.32 - 2.30i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.90iT - 7T^{2} \)
13 \( 1 + (-1.71 - 2.96i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.44 - 0.835i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.68 - 4.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.17 - 3.76i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.67iT - 31T^{2} \)
37 \( 1 + 9.48iT - 37T^{2} \)
41 \( 1 + (-2.19 + 3.80i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.74 - 4.47i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.22 - 7.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.20 - 0.694i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.3 + 6.56i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.36 - 2.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.839 - 0.484i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.2 - 5.94i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.54 + 0.892i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.01 + 6.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.39iT - 83T^{2} \)
89 \( 1 + (-9.50 + 5.48i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.18 - 1.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.32007750302114857728141745244, −10.81380637811810560072716838951, −9.374980463517585285595389689983, −9.037892126449480566274626130406, −7.67089691299399996600585350222, −6.51798142093984502901296350826, −5.62953188540060516042441168587, −4.24171606327902810409476256583, −3.13361905811187198280697605331, −2.24883822157748033209588096750, 0.72681399534773922566077661435, 3.14924334010642456394326045783, 4.30214761867371604303765536245, 5.11741185204126283335293565837, 6.28670527022467169571726689032, 7.57878783462599183256150144506, 8.177092044920151017405064284118, 8.708821176715856369673696241361, 10.37720304280108360485977872496, 10.89419328349367733900188086693

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