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Properties

Label 2-416-104.11-c1-0-0
Degree $2$
Conductor $416$
Sign $-0.836 - 0.547i$
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  + (0.928 + 1.60i)3-s + (−1.51 + 1.51i)5-s + (−2.97 + 0.797i)7-s + (−0.223 + 0.386i)9-s + (−0.865 − 0.231i)11-s + (0.159 + 3.60i)13-s + (−3.84 − 1.03i)15-s + (1.05 + 0.610i)17-s + (−6.68 + 1.79i)19-s + (−4.04 − 4.04i)21-s + (0.433 + 0.751i)23-s + 0.390i·25-s + 4.74·27-s + (−3.26 + 1.88i)29-s + (5.06 − 5.06i)31-s + ⋯
L(s)  = 1  + (0.535 + 0.928i)3-s + (−0.678 + 0.678i)5-s + (−1.12 + 0.301i)7-s + (−0.0743 + 0.128i)9-s + (−0.260 − 0.0699i)11-s + (0.0442 + 0.999i)13-s + (−0.994 − 0.266i)15-s + (0.256 + 0.147i)17-s + (−1.53 + 0.410i)19-s + (−0.883 − 0.883i)21-s + (0.0904 + 0.156i)23-s + 0.0780i·25-s + 0.912·27-s + (−0.606 + 0.350i)29-s + (0.909 − 0.909i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.836 - 0.547i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.836 - 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.282816 + 0.949214i\)
\(L(\frac12)\) \(\approx\) \(0.282816 + 0.949214i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.159 - 3.60i)T \)
good3 \( 1 + (-0.928 - 1.60i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.51 - 1.51i)T - 5iT^{2} \)
7 \( 1 + (2.97 - 0.797i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.865 + 0.231i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.05 - 0.610i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.68 - 1.79i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.433 - 0.751i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.26 - 1.88i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.06 + 5.06i)T - 31iT^{2} \)
37 \( 1 + (-2.52 + 9.43i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.12 - 11.6i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.22 - 2.44i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.24 - 4.24i)T + 47iT^{2} \)
53 \( 1 - 2.16iT - 53T^{2} \)
59 \( 1 + (0.0382 + 0.142i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-7.26 - 4.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.422 - 1.57i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.27 - 15.9i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-9.55 + 9.55i)T - 73iT^{2} \)
79 \( 1 + 6.37iT - 79T^{2} \)
83 \( 1 + (3.53 + 3.53i)T + 83iT^{2} \)
89 \( 1 + (-9.33 - 2.50i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (10.6 - 2.86i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.38764246548789437148579454738, −10.58533010989516777108197337417, −9.679217058099894734141582734341, −9.094294992364563831452115053389, −8.020398560411429931563490701016, −6.85815375726010592225065996246, −6.03217806675435271001289001740, −4.33527791755000266055799893590, −3.66432876481408207783649804122, −2.62037520448322280606885243100, 0.58425726421003469668418697927, 2.44476016849574478644300608231, 3.65919645881682509264509302341, 4.93613017296035212807327182854, 6.36772196808951909708347015785, 7.18276507894847054547160498531, 8.161556968089672830096835902865, 8.661068699279373549014132692137, 9.971210224866643484570630443669, 10.74682628725668672945377490983

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