Properties

Degree $2$
Conductor $63$
Sign $0.577 + 0.816i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.66i·2-s + 5.22·4-s + 18.5·7-s − 22.0i·8-s − 29.3i·11-s − 30.8i·14-s + 5.16·16-s − 48.9·22-s + 181. i·23-s − 125·25-s + 96.8·28-s + 304. i·29-s − 184. i·32-s − 10.5·37-s − 534.·43-s − 153. i·44-s + ⋯
L(s)  = 1  − 0.588i·2-s + 0.653·4-s + 0.999·7-s − 0.973i·8-s − 0.805i·11-s − 0.588i·14-s + 0.0807·16-s − 0.473·22-s + 1.64i·23-s − 25-s + 0.653·28-s + 1.94i·29-s − 1.02i·32-s − 0.0470·37-s − 1.89·43-s − 0.526i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.577 + 0.816i$
Motivic weight: \(3\)
Character: $\chi_{63} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.57055 - 0.812979i\)
\(L(\frac12)\) \(\approx\) \(1.57055 - 0.812979i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - 18.5T \)
good2 \( 1 + 1.66iT - 8T^{2} \)
5 \( 1 + 125T^{2} \)
11 \( 1 + 29.3iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 - 181. iT - 1.21e4T^{2} \)
29 \( 1 - 304. iT - 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 + 10.5T + 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 + 534.T + 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 + 768. iT - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 740T + 3.00e5T^{2} \)
71 \( 1 - 1.17e3iT - 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 + 1.38e3T + 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−14.26256220283879546309429296296, −13.06268613193162067845853338748, −11.68050079103781596744767865314, −11.18666223618144634064830519327, −9.944500625117449408980862581083, −8.380217848138606239718806710845, −7.08244716845334557569672896844, −5.45857681032163956187481676432, −3.47829341083152219219793015352, −1.61031405005102637955121769224, 2.14342547010408960919208194569, 4.63213618137427135380497545344, 6.13028116741244000476307604241, 7.45232098653198950654503520588, 8.389119888755360465783414245895, 10.13092149744954092187930118706, 11.33809688776693704599685285698, 12.22685079257725622210120301119, 13.83561963229848744873409908874, 14.88582848869473246378888779910

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