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Properties

Label 2-656-41.5-c1-0-3
Degree $2$
Conductor $656$
Sign $0.989 - 0.142i$
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  + (−2.22 − 2.22i)3-s + (−2.57 − 0.837i)5-s + (0.252 + 1.59i)7-s + 6.93i·9-s + (−0.314 + 0.617i)11-s + (−1.50 − 0.238i)13-s + (3.87 + 7.60i)15-s + (0.942 + 0.480i)17-s + (0.126 − 0.0201i)19-s + (2.98 − 4.11i)21-s + (−1.52 + 1.10i)23-s + (1.89 + 1.37i)25-s + (8.76 − 8.76i)27-s + (7.39 − 3.76i)29-s + (−0.373 − 1.14i)31-s + ⋯
L(s)  = 1  + (−1.28 − 1.28i)3-s + (−1.15 − 0.374i)5-s + (0.0953 + 0.602i)7-s + 2.31i·9-s + (−0.0948 + 0.186i)11-s + (−0.418 − 0.0662i)13-s + (1.00 + 1.96i)15-s + (0.228 + 0.116i)17-s + (0.0291 − 0.00461i)19-s + (0.651 − 0.897i)21-s + (−0.317 + 0.230i)23-s + (0.378 + 0.275i)25-s + (1.68 − 1.68i)27-s + (1.37 − 0.699i)29-s + (−0.0670 − 0.206i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $0.989 - 0.142i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ 0.989 - 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.527616 + 0.0377351i\)
\(L(\frac12)\) \(\approx\) \(0.527616 + 0.0377351i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.96 + 2.31i)T \)
good3 \( 1 + (2.22 + 2.22i)T + 3iT^{2} \)
5 \( 1 + (2.57 + 0.837i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.252 - 1.59i)T + (-6.65 + 2.16i)T^{2} \)
11 \( 1 + (0.314 - 0.617i)T + (-6.46 - 8.89i)T^{2} \)
13 \( 1 + (1.50 + 0.238i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.942 - 0.480i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.126 + 0.0201i)T + (18.0 - 5.87i)T^{2} \)
23 \( 1 + (1.52 - 1.10i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-7.39 + 3.76i)T + (17.0 - 23.4i)T^{2} \)
31 \( 1 + (0.373 + 1.14i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.86 - 5.75i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-6.21 - 8.55i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (-0.991 + 6.26i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (11.1 - 5.67i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-6.61 + 4.80i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.54 - 2.13i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.85 - 7.56i)T + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (1.94 - 3.82i)T + (-41.7 - 57.4i)T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 + (-8.14 - 8.14i)T + 79iT^{2} \)
83 \( 1 + 6.49T + 83T^{2} \)
89 \( 1 + (-0.451 - 2.84i)T + (-84.6 + 27.5i)T^{2} \)
97 \( 1 + (-0.548 - 1.07i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.95918250400750256769348500136, −9.842416524175380858252363981066, −8.385240781388897837875750592960, −7.84893145439152692074102351849, −7.03505828510377747810448295758, −6.10827799347400237570216231304, −5.24117345578189676315611863855, −4.29631125197284754418095370780, −2.47690698080120748852718069997, −0.955057441492730461052036673061, 0.45523990000888105098741174683, 3.26965409336938359492348279018, 4.15091017926305617540942870865, 4.81003467474177983381910094001, 5.87954510605964599060806299214, 6.89985822210511223036603592112, 7.76831881257254790361109412963, 8.999670830979960728081164948856, 9.952828782784336718700828366414, 10.79060015702435233032521145475

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