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Properties

Label 2-63-63.20-c3-0-10
Degree $2$
Conductor $63$
Sign $0.564 - 0.825i$
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  + (4.27 + 2.46i)2-s + (−3.18 − 4.10i)3-s + (8.16 + 14.1i)4-s + (7.65 + 13.2i)5-s + (−3.49 − 25.3i)6-s + (11.9 − 14.1i)7-s + 41.0i·8-s + (−6.67 + 26.1i)9-s + 75.5i·10-s + (−36.9 − 21.3i)11-s + (31.9 − 78.5i)12-s + (24.2 − 14.0i)13-s + (85.8 − 31.2i)14-s + (30.0 − 73.6i)15-s + (−35.9 + 62.2i)16-s − 82.0·17-s + ⋯
L(s)  = 1  + (1.51 + 0.871i)2-s + (−0.613 − 0.789i)3-s + (1.02 + 1.76i)4-s + (0.684 + 1.18i)5-s + (−0.237 − 1.72i)6-s + (0.643 − 0.765i)7-s + 1.81i·8-s + (−0.247 + 0.968i)9-s + 2.38i·10-s + (−1.01 − 0.584i)11-s + (0.769 − 1.89i)12-s + (0.517 − 0.299i)13-s + (1.63 − 0.595i)14-s + (0.516 − 1.26i)15-s + (−0.562 + 0.973i)16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.564 - 0.825i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.46448 + 1.30115i\)
\(L(\frac12)\) \(\approx\) \(2.46448 + 1.30115i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.18 + 4.10i)T \)
7 \( 1 + (-11.9 + 14.1i)T \)
good2 \( 1 + (-4.27 - 2.46i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-7.65 - 13.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (36.9 + 21.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-24.2 + 14.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 82.0T + 4.91e3T^{2} \)
19 \( 1 + 113. iT - 6.85e3T^{2} \)
23 \( 1 + (25.2 - 14.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-16.5 - 9.56i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (97.0 - 56.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 69.2T + 5.06e4T^{2} \)
41 \( 1 + (-242. - 419. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (194. - 337. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-53.4 + 92.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 207. iT - 1.48e5T^{2} \)
59 \( 1 + (-97.8 - 169. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (446. + 257. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-221. - 383. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 640. iT - 3.57e5T^{2} \)
73 \( 1 + 528. iT - 3.89e5T^{2} \)
79 \( 1 + (-20.7 + 35.8i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (554. - 960. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 121.T + 7.04e5T^{2} \)
97 \( 1 + (-748. - 432. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.34833148140170147761618082518, −13.46576882563791866665446443399, −13.13399771292221876958053493569, −11.32194588687220806420890360487, −10.76508560533716796001735240621, −7.914124923899236024080397435870, −6.89952232170233268415612586831, −6.08869638572709291496912500502, −4.82598907341100066600582316692, −2.74518767362978269423335835105, 1.98477795058996333056536334554, 4.21113838901891589313839751084, 5.19834389647412759200892073831, 5.88561792738594004568814422661, 8.803282729428865407510728266141, 10.13012422049428369828685175617, 11.18834266542377118348757207325, 12.25068921201734249786822375338, 12.90029168248244152220758409433, 14.10249099836935436150200185877

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