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Properties

Label 2-416-104.101-c1-0-3
Degree $2$
Conductor $416$
Sign $0.494 - 0.869i$
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  + (0.633 − 0.366i)3-s − 3.73·5-s + (3 + 1.73i)7-s + (−1.23 + 2.13i)9-s + (1 + 1.73i)11-s + (2.59 + 2.5i)13-s + (−2.36 + 1.36i)15-s + (0.232 − 0.401i)17-s + (0.633 − 1.09i)19-s + 2.53·21-s + (4.09 + 7.09i)23-s + 8.92·25-s + 4i·27-s + (−2.59 + 1.5i)29-s − 4.73i·31-s + ⋯
L(s)  = 1  + (0.366 − 0.211i)3-s − 1.66·5-s + (1.13 + 0.654i)7-s + (−0.410 + 0.711i)9-s + (0.301 + 0.522i)11-s + (0.720 + 0.693i)13-s + (−0.610 + 0.352i)15-s + (0.0562 − 0.0974i)17-s + (0.145 − 0.251i)19-s + 0.553·21-s + (0.854 + 1.48i)23-s + 1.78·25-s + 0.769i·27-s + (−0.482 + 0.278i)29-s − 0.849i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.494 - 0.869i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.494 - 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05863 + 0.615738i\)
\(L(\frac12)\) \(\approx\) \(1.05863 + 0.615738i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-2.59 - 2.5i)T \)
good3 \( 1 + (-0.633 + 0.366i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.73T + 5T^{2} \)
7 \( 1 + (-3 - 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.232 + 0.401i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.633 + 1.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.09 - 7.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.59 - 1.5i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.73iT - 31T^{2} \)
37 \( 1 + (2.13 + 3.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.96 - 4.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.19 - 1.26i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.73iT - 47T^{2} \)
53 \( 1 - 3.92iT - 53T^{2} \)
59 \( 1 + (-0.267 + 0.464i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.63 - 6.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.02 + 4.63i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 1.46T + 83T^{2} \)
89 \( 1 + (-6.46 + 3.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.19 + 3i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.53095138414858035968351549926, −10.87898092760640178407617451408, −9.197515832800522522815660832048, −8.478199150710469835580742174991, −7.75753537813985395187038211259, −7.08363827205871246387422671582, −5.39520881538272844577806825693, −4.46328523617981775878466237282, −3.34666221965466076857434735553, −1.77229169983660165144111894884, 0.834241500126876738541077878787, 3.23972986631468336240797464191, 3.93894145141199917524762921325, 4.96166744930197734887152398412, 6.50542897445817711724316051589, 7.60792092979362550038002896221, 8.361923976459178065102677400411, 8.811722598968633518589774749631, 10.46776492681225738155862611562, 11.10392593433307053345863631326

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