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Properties

Label 2-63-9.4-c3-0-9
Degree $2$
Conductor $63$
Sign $-0.425 - 0.905i$
Analytic cond. $3.71712$
Root an. cond. $1.92798$
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  + (2.80 + 4.86i)2-s + (5.15 − 0.685i)3-s + (−11.7 + 20.3i)4-s + (5.31 − 9.20i)5-s + (17.7 + 23.1i)6-s + (−3.5 − 6.06i)7-s − 87.0·8-s + (26.0 − 7.05i)9-s + 59.6·10-s + (−6.90 − 11.9i)11-s + (−46.6 + 112. i)12-s + (6.55 − 11.3i)13-s + (19.6 − 34.0i)14-s + (21.0 − 51.0i)15-s + (−150. − 260. i)16-s + 23.0·17-s + ⋯
L(s)  = 1  + (0.992 + 1.71i)2-s + (0.991 − 0.131i)3-s + (−1.46 + 2.54i)4-s + (0.475 − 0.823i)5-s + (1.21 + 1.57i)6-s + (−0.188 − 0.327i)7-s − 3.84·8-s + (0.965 − 0.261i)9-s + 1.88·10-s + (−0.189 − 0.327i)11-s + (−1.12 + 2.71i)12-s + (0.139 − 0.242i)13-s + (0.375 − 0.649i)14-s + (0.362 − 0.878i)15-s + (−2.35 − 4.07i)16-s + 0.328·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.425 - 0.905i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.425 - 0.905i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.48495 + 2.33776i\)
\(L(\frac12)\) \(\approx\) \(1.48495 + 2.33776i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.15 + 0.685i)T \)
7 \( 1 + (3.5 + 6.06i)T \)
good2 \( 1 + (-2.80 - 4.86i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-5.31 + 9.20i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (6.90 + 11.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-6.55 + 11.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 23.0T + 4.91e3T^{2} \)
19 \( 1 + 46.1T + 6.85e3T^{2} \)
23 \( 1 + (63.9 - 110. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-54.4 - 94.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-72.7 + 125. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 12.2T + 5.06e4T^{2} \)
41 \( 1 + (155. - 269. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (78.5 + 136. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (38.4 + 66.5i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 221.T + 1.48e5T^{2} \)
59 \( 1 + (243. - 421. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (465. + 806. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-111. + 192. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 336.T + 3.57e5T^{2} \)
73 \( 1 + 119.T + 3.89e5T^{2} \)
79 \( 1 + (11.5 + 19.9i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-330. - 573. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + (-850. - 1.47e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.79613721599865611042731063689, −13.71066821572039703950805555454, −13.26488159399305167017576494827, −12.31441408212157627881269502709, −9.520298428405472394147450689575, −8.495294813753999167443814072237, −7.60673394806063482665461486363, −6.23919913105924028372300242874, −4.85668954749365926491181739950, −3.45314056085812467074564592341, 2.07661515656162415773604163482, 3.10216660614921116089482640976, 4.53108657572287220260140919192, 6.31210593293804683942342069258, 8.789116225926187453281537847861, 10.01127760540784757314696924598, 10.54929843083521139349543796887, 12.04804391437161687388725528154, 13.01126118055795000214111456805, 14.02427107297266536830668390718

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