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Properties

Label 2-656-41.40-c3-0-46
Degree $2$
Conductor $656$
Sign $-0.156 + 0.987i$
Analytic cond. $38.7052$
Root an. cond. $6.22135$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.64i·3-s + 13.8·5-s + 4.28i·7-s − 4.84·9-s − 34.5i·11-s + 34.3i·13-s − 78.1i·15-s + 5.45i·17-s − 49.0i·19-s + 24.1·21-s + 27.6·23-s + 66.6·25-s − 125. i·27-s − 187. i·29-s + 52.3·31-s + ⋯
L(s)  = 1  − 1.08i·3-s + 1.23·5-s + 0.231i·7-s − 0.179·9-s − 0.946i·11-s + 0.732i·13-s − 1.34i·15-s + 0.0777i·17-s − 0.592i·19-s + 0.251·21-s + 0.251·23-s + 0.533·25-s − 0.891i·27-s − 1.20i·29-s + 0.303·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(38.7052\)
Root analytic conductor: \(6.22135\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :3/2),\ -0.156 + 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.573407527\)
\(L(\frac12)\) \(\approx\) \(2.573407527\)
\(L(\frac{5}{2})\) 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 \( 1 + (-41 + 259. i)T \)
good3 \( 1 + 5.64iT - 27T^{2} \)
5 \( 1 - 13.8T + 125T^{2} \)
7 \( 1 - 4.28iT - 343T^{2} \)
11 \( 1 + 34.5iT - 1.33e3T^{2} \)
13 \( 1 - 34.3iT - 2.19e3T^{2} \)
17 \( 1 - 5.45iT - 4.91e3T^{2} \)
19 \( 1 + 49.0iT - 6.85e3T^{2} \)
23 \( 1 - 27.6T + 1.21e4T^{2} \)
29 \( 1 + 187. iT - 2.43e4T^{2} \)
31 \( 1 - 52.3T + 2.97e4T^{2} \)
37 \( 1 - 258.T + 5.06e4T^{2} \)
43 \( 1 - 93.2T + 7.95e4T^{2} \)
47 \( 1 - 422. iT - 1.03e5T^{2} \)
53 \( 1 + 105. iT - 1.48e5T^{2} \)
59 \( 1 + 529.T + 2.05e5T^{2} \)
61 \( 1 - 428.T + 2.26e5T^{2} \)
67 \( 1 + 932. iT - 3.00e5T^{2} \)
71 \( 1 - 702. iT - 3.57e5T^{2} \)
73 \( 1 + 974.T + 3.89e5T^{2} \)
79 \( 1 - 205. iT - 4.93e5T^{2} \)
83 \( 1 - 1.09e3T + 5.71e5T^{2} \)
89 \( 1 + 1.50e3iT - 7.04e5T^{2} \)
97 \( 1 + 293. iT - 9.12e5T^{2} \)
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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

−9.743810117882657319474319788458, −9.082350320563709179563485736772, −8.108414245644195169954579515000, −7.12988621959149804621129978529, −6.22549329212932806477665564753, −5.77315169116710652012417298207, −4.38041736513890834083987301595, −2.72977587064162604336120877423, −1.88410596469716697136893465126, −0.76683278785903561705210376217, 1.37570493160413956768436388296, 2.68079119949212684960867118166, 3.93454730030472708522167773858, 4.92217988989510771409041544368, 5.64430217695878146672041881159, 6.72847497936210141159231628638, 7.77697709132176350414808787312, 9.034305704110278681190255271389, 9.661118375327652308023526562548, 10.28376480073195890941466759660

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