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Properties

Label 2-63-7.2-c9-0-14
Degree $2$
Conductor $63$
Sign $0.386 + 0.922i$
Analytic cond. $32.4472$
Root an. cond. $5.69624$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (19.4 − 33.7i)2-s + (−502. − 871. i)4-s + (−1.23e3 + 2.14e3i)5-s + (6.00e3 − 2.08e3i)7-s − 1.92e4·8-s + (4.82e4 + 8.35e4i)10-s + (9.74e3 + 1.68e4i)11-s + 2.30e4·13-s + (4.67e4 − 2.43e5i)14-s + (−1.17e5 + 2.03e5i)16-s + (2.82e5 + 4.88e5i)17-s + (2.59e5 − 4.49e5i)19-s + 2.49e6·20-s + 7.59e5·22-s + (7.72e5 − 1.33e6i)23-s + ⋯
L(s)  = 1  + (0.860 − 1.49i)2-s + (−0.982 − 1.70i)4-s + (−0.886 + 1.53i)5-s + (0.944 − 0.327i)7-s − 1.66·8-s + (1.52 + 2.64i)10-s + (0.200 + 0.347i)11-s + 0.224·13-s + (0.325 − 1.69i)14-s + (−0.447 + 0.775i)16-s + (0.818 + 1.41i)17-s + (0.457 − 0.791i)19-s + 3.48·20-s + 0.690·22-s + (0.575 − 0.997i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(32.4472\)
Root analytic conductor: \(5.69624\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :9/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.50162 - 1.66379i\)
\(L(\frac12)\) \(\approx\) \(2.50162 - 1.66379i\)
\(L(\frac{11}{2})\) 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 \( 1 + (-6.00e3 + 2.08e3i)T \)
good2 \( 1 + (-19.4 + 33.7i)T + (-256 - 443. i)T^{2} \)
5 \( 1 + (1.23e3 - 2.14e3i)T + (-9.76e5 - 1.69e6i)T^{2} \)
11 \( 1 + (-9.74e3 - 1.68e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 - 2.30e4T + 1.06e10T^{2} \)
17 \( 1 + (-2.82e5 - 4.88e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-2.59e5 + 4.49e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (-7.72e5 + 1.33e6i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 - 4.76e6T + 1.45e13T^{2} \)
31 \( 1 + (-2.35e6 - 4.08e6i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 + (9.94e6 - 1.72e7i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 - 1.93e7T + 3.27e14T^{2} \)
43 \( 1 - 9.57e5T + 5.02e14T^{2} \)
47 \( 1 + (1.71e7 - 2.97e7i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (-1.01e7 - 1.75e7i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (6.21e6 + 1.07e7i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (-8.00e7 + 1.38e8i)T + (-5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (-3.39e7 - 5.88e7i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 - 2.84e8T + 4.58e16T^{2} \)
73 \( 1 + (-1.30e8 - 2.26e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (4.18e7 - 7.25e7i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + 1.69e7T + 1.86e17T^{2} \)
89 \( 1 + (-5.07e7 + 8.78e7i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 - 8.99e8T + 7.60e17T^{2} \)
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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

−12.53057445015252983121739446682, −11.60233135626335828723701267336, −10.82898641070659082680249477441, −10.20876934652773416366927665335, −8.127561600084179093342251201393, −6.66190219131647015316276787398, −4.76093235905677142088362011175, −3.67087886334730060646385634896, −2.65206715669405073593106356705, −1.12800689203215577597028500405, 0.914910810276086381483152321107, 3.77636222587912474958590271424, 4.92715911574459564162440570211, 5.55685848509580674421774426140, 7.44956207370274259754484330149, 8.157464275081787314992244236692, 9.115309302675675713787944072645, 11.64210304208516003529320861155, 12.32982951215497133249809720584, 13.54678716604371559666046169731

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