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Properties

Label 2-656-41.20-c1-0-0
Degree $2$
Conductor $656$
Sign $-0.477 + 0.878i$
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  + (−0.964 + 0.964i)3-s + (−2.03 + 2.80i)5-s + (−1.29 − 0.661i)7-s + 1.13i·9-s + (−0.285 + 1.80i)11-s + (−2.91 − 5.71i)13-s + (−0.739 − 4.67i)15-s + (3.14 + 0.497i)17-s + (−1.51 + 2.98i)19-s + (1.89 − 0.614i)21-s + (1.67 − 5.15i)23-s + (−2.16 − 6.66i)25-s + (−3.99 − 3.99i)27-s + (−0.335 + 0.0530i)29-s + (3.04 − 2.21i)31-s + ⋯
L(s)  = 1  + (−0.557 + 0.557i)3-s + (−0.911 + 1.25i)5-s + (−0.490 − 0.249i)7-s + 0.379i·9-s + (−0.0861 + 0.544i)11-s + (−0.807 − 1.58i)13-s + (−0.191 − 1.20i)15-s + (0.762 + 0.120i)17-s + (−0.348 + 0.683i)19-s + (0.412 − 0.134i)21-s + (0.349 − 1.07i)23-s + (−0.433 − 1.33i)25-s + (−0.768 − 0.768i)27-s + (−0.0622 + 0.00985i)29-s + (0.546 − 0.396i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $-0.477 + 0.878i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ -0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0207925 - 0.0349677i\)
\(L(\frac12)\) \(\approx\) \(0.0207925 - 0.0349677i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.22 + 1.51i)T \)
good3 \( 1 + (0.964 - 0.964i)T - 3iT^{2} \)
5 \( 1 + (2.03 - 2.80i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.29 + 0.661i)T + (4.11 + 5.66i)T^{2} \)
11 \( 1 + (0.285 - 1.80i)T + (-10.4 - 3.39i)T^{2} \)
13 \( 1 + (2.91 + 5.71i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-3.14 - 0.497i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (1.51 - 2.98i)T + (-11.1 - 15.3i)T^{2} \)
23 \( 1 + (-1.67 + 5.15i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (0.335 - 0.0530i)T + (27.5 - 8.96i)T^{2} \)
31 \( 1 + (-3.04 + 2.21i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.993 - 0.721i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (7.02 + 2.28i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (-1.40 + 0.717i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-1.12 + 0.178i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (4.16 - 12.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.39 - 1.42i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (1.04 + 6.58i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (-1.29 + 8.16i)T + (-67.5 - 21.9i)T^{2} \)
73 \( 1 - 8.99iT - 73T^{2} \)
79 \( 1 + (7.20 - 7.20i)T - 79iT^{2} \)
83 \( 1 - 7.11T + 83T^{2} \)
89 \( 1 + (12.9 + 6.58i)T + (52.3 + 72.0i)T^{2} \)
97 \( 1 + (-0.514 - 3.24i)T + (-92.2 + 29.9i)T^{2} \)
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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

−10.84571912536949751293200429124, −10.25935939615586380643765974368, −9.998152261206441447693380875072, −8.227484543478751543704824723036, −7.60158639084641279056786608964, −6.76924595779183588459996464493, −5.68555176642582253138139377549, −4.66223354836631764168601691151, −3.56898896683640301130096624041, −2.65757661131387864656638178221, 0.02432679288620762356945125529, 1.40887361827248609313598251931, 3.30473981402404062236988034396, 4.47232098842185026300114355905, 5.32627008062215288459120827669, 6.45857770277536896817998418339, 7.22231951913983174514190728675, 8.230869038505093847632269346447, 9.130194324917163943253096248240, 9.665810094579281370300833449438

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