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Properties

Label 2-416-416.77-c1-0-11
Degree $2$
Conductor $416$
Sign $0.580 - 0.814i$
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  + (−1.31 + 0.510i)2-s + (0.324 − 0.782i)3-s + (1.47 − 1.34i)4-s + (0.174 + 0.420i)5-s + (−0.0279 + 1.19i)6-s + (2.63 + 2.63i)7-s + (−1.26 + 2.53i)8-s + (1.61 + 1.61i)9-s + (−0.444 − 0.465i)10-s + (−2.03 + 0.842i)11-s + (−0.574 − 1.59i)12-s + (−2.76 − 2.31i)13-s + (−4.81 − 2.12i)14-s + 0.385·15-s + (0.373 − 3.98i)16-s + 4.77i·17-s + ⋯
L(s)  = 1  + (−0.932 + 0.360i)2-s + (0.187 − 0.451i)3-s + (0.739 − 0.673i)4-s + (0.0778 + 0.187i)5-s + (−0.0114 + 0.488i)6-s + (0.995 + 0.995i)7-s + (−0.446 + 0.894i)8-s + (0.538 + 0.538i)9-s + (−0.140 − 0.147i)10-s + (−0.613 + 0.254i)11-s + (−0.165 − 0.459i)12-s + (−0.765 − 0.643i)13-s + (−1.28 − 0.568i)14-s + 0.0994·15-s + (0.0933 − 0.995i)16-s + 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.580 - 0.814i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.932958 + 0.480560i\)
\(L(\frac12)\) \(\approx\) \(0.932958 + 0.480560i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.510i)T \)
13 \( 1 + (2.76 + 2.31i)T \)
good3 \( 1 + (-0.324 + 0.782i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.174 - 0.420i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.63 - 2.63i)T + 7iT^{2} \)
11 \( 1 + (2.03 - 0.842i)T + (7.77 - 7.77i)T^{2} \)
17 \( 1 - 4.77iT - 17T^{2} \)
19 \( 1 + (0.598 - 1.44i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-3.12 - 3.12i)T + 23iT^{2} \)
29 \( 1 + (1.02 - 2.47i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.90iT - 31T^{2} \)
37 \( 1 + (2.32 + 5.61i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-2.95 + 2.95i)T - 41iT^{2} \)
43 \( 1 + (-4.00 - 9.66i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 1.36T + 47T^{2} \)
53 \( 1 + (0.464 + 1.12i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.498 - 1.20i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.55 + 8.57i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-7.70 - 3.19i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (9.25 + 9.25i)T + 71iT^{2} \)
73 \( 1 + (-10.4 + 10.4i)T - 73iT^{2} \)
79 \( 1 + 1.42iT - 79T^{2} \)
83 \( 1 + (4.61 - 11.1i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.78 + 5.78i)T + 89iT^{2} \)
97 \( 1 + 9.17iT - 97T^{2} \)
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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

−10.98400193997475920133298165407, −10.50095928790751261219471866027, −9.422933151520359080559142677395, −8.330529324677037419726373152451, −7.87673964788035726621938233810, −6.98324414203059736353295124834, −5.71603885271011729772341609745, −4.90712649601190569219547259059, −2.60115287358147544115837895481, −1.67500491566344341712099176199, 0.976107232010067268318660425407, 2.61623853066820194243635591961, 4.05793620258765451436399181766, 4.99768094966376238468077017643, 6.90254466857108773226776792193, 7.39641088510646886575445938060, 8.519111416202564210592424520651, 9.351796731429093648189992587138, 10.12209437912093059011656228234, 10.92095638208681322079427104263

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