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Properties

Label 2-656-41.36-c1-0-4
Degree $2$
Conductor $656$
Sign $0.221 - 0.975i$
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  + (0.604 − 0.604i)3-s + (−3.27 − 1.06i)5-s + (−1.70 + 0.270i)7-s + 2.26i·9-s + (3.99 + 2.03i)11-s + (−0.380 + 2.40i)13-s + (−2.62 + 1.33i)15-s + (−0.138 + 0.272i)17-s + (−0.274 − 1.73i)19-s + (−0.867 + 1.19i)21-s + (−3.75 + 2.72i)23-s + (5.57 + 4.05i)25-s + (3.18 + 3.18i)27-s + (4.39 + 8.62i)29-s + (2.28 + 7.03i)31-s + ⋯
L(s)  = 1  + (0.348 − 0.348i)3-s + (−1.46 − 0.476i)5-s + (−0.644 + 0.102i)7-s + 0.756i·9-s + (1.20 + 0.613i)11-s + (−0.105 + 0.665i)13-s + (−0.678 + 0.345i)15-s + (−0.0336 + 0.0660i)17-s + (−0.0629 − 0.397i)19-s + (−0.189 + 0.260i)21-s + (−0.782 + 0.568i)23-s + (1.11 + 0.810i)25-s + (0.612 + 0.612i)27-s + (0.815 + 1.60i)29-s + (0.410 + 1.26i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $0.221 - 0.975i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ 0.221 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709259 + 0.566278i\)
\(L(\frac12)\) \(\approx\) \(0.709259 + 0.566278i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.99 - 2.26i)T \)
good3 \( 1 + (-0.604 + 0.604i)T - 3iT^{2} \)
5 \( 1 + (3.27 + 1.06i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (1.70 - 0.270i)T + (6.65 - 2.16i)T^{2} \)
11 \( 1 + (-3.99 - 2.03i)T + (6.46 + 8.89i)T^{2} \)
13 \( 1 + (0.380 - 2.40i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (0.138 - 0.272i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (0.274 + 1.73i)T + (-18.0 + 5.87i)T^{2} \)
23 \( 1 + (3.75 - 2.72i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-4.39 - 8.62i)T + (-17.0 + 23.4i)T^{2} \)
31 \( 1 + (-2.28 - 7.03i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.28 + 7.02i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-1.53 - 2.10i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (6.25 + 0.991i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (0.556 + 1.09i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (2.66 - 1.93i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.655 - 0.901i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-9.09 + 4.63i)T + (39.3 - 54.2i)T^{2} \)
71 \( 1 + (0.475 + 0.242i)T + (41.7 + 57.4i)T^{2} \)
73 \( 1 - 8.56iT - 73T^{2} \)
79 \( 1 + (-6.09 + 6.09i)T - 79iT^{2} \)
83 \( 1 + 9.88T + 83T^{2} \)
89 \( 1 + (14.4 - 2.28i)T + (84.6 - 27.5i)T^{2} \)
97 \( 1 + (-8.80 + 4.48i)T + (57.0 - 78.4i)T^{2} \)
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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

−10.87693871226281321579110566858, −9.693767772103886335530648187826, −8.826517672141859057530906679504, −8.167541876860024489761960331970, −7.18323987791596275547191940652, −6.64144110480176323938672310723, −4.99052400758021976272873412119, −4.16927881673713531064830598797, −3.17557772230082277207638124812, −1.55382334683831640205902321856, 0.49298128472625800407219213082, 2.94817073151948511569575342531, 3.72752106723913155294904379199, 4.32514333987413804369088186828, 6.16538572320201443651416978128, 6.68864705837227583734538891512, 7.954168580564052042429003343363, 8.435982744596099778672114935467, 9.606010489161228217248614930710, 10.19471338306858305979534444812

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