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Properties

Label 2-416-416.99-c1-0-34
Degree $2$
Conductor $416$
Sign $0.194 + 0.980i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
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  + (−1.29 − 0.572i)2-s + (0.174 − 0.420i)3-s + (1.34 + 1.47i)4-s + (1.36 − 3.30i)5-s + (−0.466 + 0.444i)6-s + 1.57·7-s + (−0.893 − 2.68i)8-s + (1.97 + 1.97i)9-s + (−3.66 + 3.49i)10-s + (1.42 + 0.591i)11-s + (0.857 − 0.308i)12-s + (1.31 − 3.35i)13-s + (−2.03 − 0.901i)14-s + (−1.15 − 1.15i)15-s + (−0.380 + 3.98i)16-s − 3.74·17-s + ⋯
L(s)  = 1  + (−0.914 − 0.404i)2-s + (0.100 − 0.242i)3-s + (0.672 + 0.739i)4-s + (0.612 − 1.47i)5-s + (−0.190 + 0.181i)6-s + 0.595·7-s + (−0.315 − 0.948i)8-s + (0.658 + 0.658i)9-s + (−1.15 + 1.10i)10-s + (0.430 + 0.178i)11-s + (0.247 − 0.0889i)12-s + (0.364 − 0.931i)13-s + (−0.544 − 0.240i)14-s + (−0.297 − 0.297i)15-s + (−0.0950 + 0.995i)16-s − 0.908·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.194 + 0.980i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.194 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.912266 - 0.749503i\)
\(L(\frac12)\) \(\approx\) \(0.912266 - 0.749503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.572i)T \)
13 \( 1 + (-1.31 + 3.35i)T \)
good3 \( 1 + (-0.174 + 0.420i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.36 + 3.30i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 - 1.57T + 7T^{2} \)
11 \( 1 + (-1.42 - 0.591i)T + (7.77 + 7.77i)T^{2} \)
17 \( 1 + 3.74T + 17T^{2} \)
19 \( 1 + (-3.19 - 7.70i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (3.88 + 3.88i)T + 23iT^{2} \)
29 \( 1 + (-3.22 - 1.33i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.41 + 1.41i)T - 31iT^{2} \)
37 \( 1 + (1.01 + 0.418i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + 10.5iT - 41T^{2} \)
43 \( 1 + (0.478 - 0.198i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + (1.47 - 1.47i)T - 47iT^{2} \)
53 \( 1 + (4.69 - 1.94i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-2.03 + 4.91i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (6.76 + 2.80i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (5.79 - 2.40i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 - 5.97iT - 71T^{2} \)
73 \( 1 - 1.85T + 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + (-4.57 - 11.0i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 - 13.4iT - 89T^{2} \)
97 \( 1 + (-3.24 - 3.24i)T + 97iT^{2} \)
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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

−10.75831902042478367561461674736, −10.06295342010941862131969277578, −9.174539973517747352269681017196, −8.232222102657673474790338451709, −7.85831989102969014407659045764, −6.39036308384731141799716086350, −5.17795163453634087013946263402, −4.01817800644349294277184549620, −2.06315976226253288932293429825, −1.19387107390197202969420997085, 1.69491923218093880099836660849, 3.03435180236402031710270399137, 4.66920229922147613012232930768, 6.30109727936244895579977511628, 6.67683493975272077593229297155, 7.57338637556831100428449329151, 8.927344484746242294946723252071, 9.540666672059606294819202743477, 10.34692147442058192033978621603, 11.30235943110030309734721186895

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