Properties

Label 2-656-41.39-c1-0-19
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: $\chi_{656} (449, \cdot )$
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

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

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