Properties

Label 2-648-8.5-c1-0-24
Degree $2$
Conductor $648$
Sign $0.707 - 0.707i$
Analytic cond. $5.17430$
Root an. cond. $2.27471$
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  + (1 + i)2-s + 2i·4-s − 2i·5-s + 4·7-s + (−2 + 2i)8-s + (2 − 2i)10-s − 3i·11-s + 2i·13-s + (4 + 4i)14-s − 4·16-s + 5·17-s + i·19-s + 4·20-s + (3 − 3i)22-s + 2·23-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s − 0.894i·5-s + 1.51·7-s + (−0.707 + 0.707i)8-s + (0.632 − 0.632i)10-s − 0.904i·11-s + 0.554i·13-s + (1.06 + 1.06i)14-s − 16-s + 1.21·17-s + 0.229i·19-s + 0.894·20-s + (0.639 − 0.639i)22-s + 0.417·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29219 + 0.949459i\)
\(L(\frac12)\) \(\approx\) \(2.29219 + 0.949459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
good5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - 11iT - 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + iT - 59T^{2} \)
61 \( 1 + 12iT - 61T^{2} \)
67 \( 1 - 3iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−11.05676370223419007399706452406, −9.546120881916243549839023310555, −8.430571003450487946054226638599, −8.213069451344319068318173214662, −7.16912319377247514927321894021, −5.91366289011777225508440899350, −5.12522262100955812461867238592, −4.49006272121340995238732653547, −3.25282451476654786510245084972, −1.48808577906672131281369178833, 1.48659727631433953645201304522, 2.62773864332262740372489384822, 3.77047052516372296380308774509, 4.92125145007063426484753044798, 5.55590599032313466479475713771, 6.90467503324176727115680453932, 7.64329466748602418269165683526, 8.821791370928858914778684800658, 10.03853164089875178232434891928, 10.56403900665787677815629779259

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