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Properties

Label 2-656-41.8-c1-0-13
Degree $2$
Conductor $656$
Sign $0.995 + 0.0939i$
Analytic cond. $5.23818$
Root an. cond. $2.28870$
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  + (0.242 + 0.242i)3-s + (2.26 − 0.734i)5-s + (4.85 + 0.768i)7-s − 2.88i·9-s + (−1.51 + 0.773i)11-s + (0.621 + 3.92i)13-s + (0.727 + 0.370i)15-s + (−1.24 − 2.44i)17-s + (0.150 − 0.953i)19-s + (0.990 + 1.36i)21-s + (−5.46 − 3.97i)23-s + (0.530 − 0.385i)25-s + (1.42 − 1.42i)27-s + (0.230 − 0.451i)29-s + (−0.182 + 0.561i)31-s + ⋯
L(s)  = 1  + (0.140 + 0.140i)3-s + (1.01 − 0.328i)5-s + (1.83 + 0.290i)7-s − 0.960i·9-s + (−0.457 + 0.233i)11-s + (0.172 + 1.08i)13-s + (0.187 + 0.0956i)15-s + (−0.302 − 0.593i)17-s + (0.0346 − 0.218i)19-s + (0.216 + 0.297i)21-s + (−1.13 − 0.828i)23-s + (0.106 − 0.0770i)25-s + (0.274 − 0.274i)27-s + (0.0427 − 0.0839i)29-s + (−0.0327 + 0.100i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $0.995 + 0.0939i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ 0.995 + 0.0939i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13731 - 0.100659i\)
\(L(\frac12)\) \(\approx\) \(2.13731 - 0.100659i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 + (4.01 - 4.98i)T \)
good3 \( 1 + (-0.242 - 0.242i)T + 3iT^{2} \)
5 \( 1 + (-2.26 + 0.734i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-4.85 - 0.768i)T + (6.65 + 2.16i)T^{2} \)
11 \( 1 + (1.51 - 0.773i)T + (6.46 - 8.89i)T^{2} \)
13 \( 1 + (-0.621 - 3.92i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (1.24 + 2.44i)T + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.150 + 0.953i)T + (-18.0 - 5.87i)T^{2} \)
23 \( 1 + (5.46 + 3.97i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.230 + 0.451i)T + (-17.0 - 23.4i)T^{2} \)
31 \( 1 + (0.182 - 0.561i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.31 - 4.04i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (3.16 - 4.35i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (-4.96 + 0.786i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-3.46 + 6.79i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-6.81 - 4.95i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.408 - 0.562i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3.63 - 1.85i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (6.47 - 3.30i)T + (41.7 - 57.4i)T^{2} \)
73 \( 1 + 9.72iT - 73T^{2} \)
79 \( 1 + (6.15 + 6.15i)T + 79iT^{2} \)
83 \( 1 + 4.81T + 83T^{2} \)
89 \( 1 + (3.63 + 0.576i)T + (84.6 + 27.5i)T^{2} \)
97 \( 1 + (3.63 + 1.85i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.41714180415934074250313884199, −9.588698755027913073101522266501, −8.807214228039778845535892122007, −8.145946833921734325134939235684, −6.92808381116998284850968118863, −5.93258109758824279154584579889, −4.96397251272931649470798487117, −4.23261761644015491348287326498, −2.42665814454762275559804902946, −1.48514344259042214893138677282, 1.61978115031551121332273837630, 2.41092778694600272218209490839, 4.07809461017950687779187831139, 5.36091020950108393622978032251, 5.68537178226879428308516147809, 7.24499334021490317225541367037, 8.038257067927635528918817498977, 8.500226743243085752098373474716, 9.956945323626946567866782301885, 10.62231312519793457669367348852

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