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Properties

Label 2-63-63.4-c3-0-13
Degree $2$
Conductor $63$
Sign $0.607 - 0.794i$
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  + (−2.71 + 4.70i)2-s + (4.53 − 2.53i)3-s + (−10.7 − 18.6i)4-s + 13.1·5-s + (−0.382 + 28.2i)6-s + (12.4 − 13.7i)7-s + 73.2·8-s + (14.1 − 23.0i)9-s + (−35.6 + 61.6i)10-s − 18.2·11-s + (−95.9 − 57.1i)12-s + (−12.9 + 22.3i)13-s + (30.6 + 95.7i)14-s + (59.4 − 33.2i)15-s + (−112. + 195. i)16-s + (1.04 − 1.80i)17-s + ⋯
L(s)  = 1  + (−0.959 + 1.66i)2-s + (0.872 − 0.488i)3-s + (−1.34 − 2.32i)4-s + 1.17·5-s + (−0.0260 + 1.91i)6-s + (0.672 − 0.740i)7-s + 3.23·8-s + (0.523 − 0.852i)9-s + (−1.12 + 1.95i)10-s − 0.500·11-s + (−2.30 − 1.37i)12-s + (−0.275 + 0.477i)13-s + (0.585 + 1.82i)14-s + (1.02 − 0.572i)15-s + (−1.76 + 3.05i)16-s + (0.0148 − 0.0257i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.607 - 0.794i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.607 - 0.794i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.19129 + 0.588338i\)
\(L(\frac12)\) \(\approx\) \(1.19129 + 0.588338i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.53 + 2.53i)T \)
7 \( 1 + (-12.4 + 13.7i)T \)
good2 \( 1 + (2.71 - 4.70i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 13.1T + 125T^{2} \)
11 \( 1 + 18.2T + 1.33e3T^{2} \)
13 \( 1 + (12.9 - 22.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-1.04 + 1.80i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-10.2 - 17.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 68.4T + 1.21e4T^{2} \)
29 \( 1 + (-132. - 229. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-15.2 - 26.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (143. + 249. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (152. - 263. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (119. + 207. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (209. - 363. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (114. - 198. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (399. + 692. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (47.1 - 81.6i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-284. - 492. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 211.T + 3.57e5T^{2} \)
73 \( 1 + (314. - 544. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-167. + 289. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-163. - 282. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-160. - 278. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (60.9 + 105. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.41820869858095000091649888165, −14.16000865596483833084593380926, −13.12932294799433074346107587942, −10.49414587349283807735593039804, −9.536652236280952219692833219407, −8.551078364773910440189264643438, −7.46471946227157240213601920753, −6.53634053673999826077642151984, −5.01984871231602295820752153492, −1.47622568325114222751674887792, 1.90155751974991857296507162862, 2.91835219813142273948114212443, 4.89777672978159766591547260041, 7.956730566090499919183906826630, 8.856368806457064834471303428831, 9.821360387812451717847869952093, 10.49924397381039064631127079159, 11.81603477220010285715827447114, 13.11563746418378710814363820554, 13.83009623763507305323302299602

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