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Properties

Label 2-63-7.2-c3-0-8
Degree $2$
Conductor $63$
Sign $-0.817 + 0.575i$
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  + (2.65 − 4.59i)2-s + (−10.0 − 17.4i)4-s + (2.78 − 4.81i)5-s + (9.67 + 15.7i)7-s − 64.6·8-s + (−14.7 − 25.5i)10-s + (−6.95 − 12.0i)11-s + 38.6·13-s + (98.2 − 2.58i)14-s + (−90.8 + 157. i)16-s + (21.7 + 37.6i)17-s + (54.5 − 94.4i)19-s − 112.·20-s − 73.8·22-s + (−37.4 + 64.8i)23-s + ⋯
L(s)  = 1  + (0.938 − 1.62i)2-s + (−1.26 − 2.18i)4-s + (0.248 − 0.430i)5-s + (0.522 + 0.852i)7-s − 2.85·8-s + (−0.466 − 0.808i)10-s + (−0.190 − 0.330i)11-s + 0.825·13-s + (1.87 − 0.0492i)14-s + (−1.41 + 2.45i)16-s + (0.310 + 0.537i)17-s + (0.658 − 1.14i)19-s − 1.25·20-s − 0.715·22-s + (−0.339 + 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.817 + 0.575i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.817 + 0.575i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.659720 - 2.08377i\)
\(L(\frac12)\) \(\approx\) \(0.659720 - 2.08377i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.67 - 15.7i)T \)
good2 \( 1 + (-2.65 + 4.59i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-2.78 + 4.81i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (6.95 + 12.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 38.6T + 2.19e3T^{2} \)
17 \( 1 + (-21.7 - 37.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-54.5 + 94.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (37.4 - 64.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 72.3T + 2.43e4T^{2} \)
31 \( 1 + (32.0 + 55.4i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (94.3 - 163. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 24.7T + 6.89e4T^{2} \)
43 \( 1 + 243.T + 7.95e4T^{2} \)
47 \( 1 + (310. - 537. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (143. + 249. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-262. - 454. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-191. + 332. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (99.0 + 171. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 785.T + 3.57e5T^{2} \)
73 \( 1 + (-165. - 286. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (218. - 379. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 241.T + 5.71e5T^{2} \)
89 \( 1 + (-792. + 1.37e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 79.2T + 9.12e5T^{2} \)
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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

−13.59976717776953957200097538267, −12.86954786105251687120336946130, −11.72966220786294037223248221794, −11.04622734629242940918275257758, −9.683832779665303685015398886956, −8.610006958441256217182506569693, −5.83666597737119012847344002842, −4.82826556312930327921916471797, −3.13634886043002296919491938856, −1.46787699426438346279439529441, 3.67537473826489377957443694380, 5.05044062002673735222264051969, 6.38646242417868593798458354133, 7.43602404321788974473835527392, 8.449261892294487046613358139300, 10.26802147440583219631418133124, 11.96431013018856328664871605781, 13.25182595073990437206489345230, 14.13333603821986417699885421662, 14.65311548652026229514232130452

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