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Properties

Label 2-656-16.13-c1-0-18
Degree $2$
Conductor $656$
Sign $0.880 - 0.473i$
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  + (−0.508 + 1.31i)2-s + (−0.665 − 0.665i)3-s + (−1.48 − 1.34i)4-s + (−2.95 + 2.95i)5-s + (1.21 − 0.540i)6-s − 0.781i·7-s + (2.52 − 1.27i)8-s − 2.11i·9-s + (−2.39 − 5.39i)10-s + (−0.497 + 0.497i)11-s + (0.0947 + 1.87i)12-s + (−2.65 − 2.65i)13-s + (1.03 + 0.397i)14-s + 3.92·15-s + (0.402 + 3.97i)16-s + 5.16·17-s + ⋯
L(s)  = 1  + (−0.359 + 0.933i)2-s + (−0.384 − 0.384i)3-s + (−0.741 − 0.670i)4-s + (−1.31 + 1.31i)5-s + (0.496 − 0.220i)6-s − 0.295i·7-s + (0.892 − 0.451i)8-s − 0.704i·9-s + (−0.757 − 1.70i)10-s + (−0.149 + 0.149i)11-s + (0.0273 + 0.542i)12-s + (−0.735 − 0.735i)13-s + (0.275 + 0.106i)14-s + 1.01·15-s + (0.100 + 0.994i)16-s + 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $0.880 - 0.473i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ 0.880 - 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.641843 + 0.161650i\)
\(L(\frac12)\) \(\approx\) \(0.641843 + 0.161650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.508 - 1.31i)T \)
41 \( 1 - iT \)
good3 \( 1 + (0.665 + 0.665i)T + 3iT^{2} \)
5 \( 1 + (2.95 - 2.95i)T - 5iT^{2} \)
7 \( 1 + 0.781iT - 7T^{2} \)
11 \( 1 + (0.497 - 0.497i)T - 11iT^{2} \)
13 \( 1 + (2.65 + 2.65i)T + 13iT^{2} \)
17 \( 1 - 5.16T + 17T^{2} \)
19 \( 1 + (0.814 + 0.814i)T + 19iT^{2} \)
23 \( 1 - 6.09iT - 23T^{2} \)
29 \( 1 + (-1.90 - 1.90i)T + 29iT^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 + (0.128 - 0.128i)T - 37iT^{2} \)
43 \( 1 + (-2.14 + 2.14i)T - 43iT^{2} \)
47 \( 1 - 2.63T + 47T^{2} \)
53 \( 1 + (-9.40 + 9.40i)T - 53iT^{2} \)
59 \( 1 + (-4.35 + 4.35i)T - 59iT^{2} \)
61 \( 1 + (-2.58 - 2.58i)T + 61iT^{2} \)
67 \( 1 + (-1.97 - 1.97i)T + 67iT^{2} \)
71 \( 1 - 5.20iT - 71T^{2} \)
73 \( 1 + 9.21iT - 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 + (10.7 + 10.7i)T + 83iT^{2} \)
89 \( 1 + 7.19iT - 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−10.31804169684830556115078186037, −9.986054804597816496612283708245, −8.525861285156910124654474382757, −7.57928409708507194965192870332, −7.26799571781779904617855531089, −6.43592785786935729740963571274, −5.42325119099445707015380778154, −4.09106133977395318496275549342, −3.10505127303847422576803294892, −0.64741709832227570921237535389, 0.870134688729634909153934745762, 2.59485487940191376305971329099, 4.13568194396961698808815251301, 4.57673702760899100890254046702, 5.48401700019052267231126784251, 7.37812952097087766726707562755, 8.195864215698140544759887326849, 8.672581515367348396942310601525, 9.748055201026962319837550344973, 10.48133395534880495350826077258

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