Properties

Label 2-63-63.4-c3-0-7
Degree $2$
Conductor $63$
Sign $-0.583 - 0.811i$
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  + (−1.33 + 2.30i)2-s + (0.537 + 5.16i)3-s + (0.454 + 0.787i)4-s + 19.2·5-s + (−12.6 − 5.64i)6-s + (18.0 + 4.02i)7-s − 23.7·8-s + (−26.4 + 5.55i)9-s + (−25.6 + 44.3i)10-s − 38.5·11-s + (−3.82 + 2.77i)12-s + (38.2 − 66.3i)13-s + (−33.3 + 36.3i)14-s + (10.3 + 99.4i)15-s + (27.9 − 48.4i)16-s + (−11.9 + 20.6i)17-s + ⋯
L(s)  = 1  + (−0.470 + 0.815i)2-s + (0.103 + 0.994i)3-s + (0.0568 + 0.0983i)4-s + 1.72·5-s + (−0.859 − 0.383i)6-s + (0.976 + 0.217i)7-s − 1.04·8-s + (−0.978 + 0.205i)9-s + (−0.810 + 1.40i)10-s − 1.05·11-s + (−0.0919 + 0.0666i)12-s + (0.816 − 1.41i)13-s + (−0.636 + 0.693i)14-s + (0.178 + 1.71i)15-s + (0.436 − 0.756i)16-s + (−0.170 + 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 - 0.811i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.583 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.583 - 0.811i$
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: \((2,\ 63,\ (\ :3/2),\ -0.583 - 0.811i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.696475 + 1.35840i\)
\(L(\frac12)\) \(\approx\) \(0.696475 + 1.35840i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.537 - 5.16i)T \)
7 \( 1 + (-18.0 - 4.02i)T \)
good2 \( 1 + (1.33 - 2.30i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 19.2T + 125T^{2} \)
11 \( 1 + 38.5T + 1.33e3T^{2} \)
13 \( 1 + (-38.2 + 66.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (11.9 - 20.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (23.6 + 41.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 13.5T + 1.21e4T^{2} \)
29 \( 1 + (53.5 + 92.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-79.3 - 137. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-23.0 - 39.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-101. + 175. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (42.1 + 72.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (236. - 410. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (39.8 - 69.0i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-158. - 274. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-81.8 + 141. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (270. + 467. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 810.T + 3.57e5T^{2} \)
73 \( 1 + (142. - 246. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (367. - 635. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (290. + 503. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (463. + 803. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (413. + 715. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.15674509790890568067127223820, −14.04147456269357930295768875747, −12.89306018650951280378500800075, −11.00726338765178904910392179750, −10.12288899142522825209463394129, −8.883944531545785328252780930280, −8.032722863359297405888978845599, −6.02972130447035964809398669105, −5.23871886480392930746770781180, −2.69127042090109608461447388650, 1.48317083427460745636174173350, 2.31627856685167987067369783773, 5.52711756318717106041746615358, 6.61670041383751173128662571741, 8.422297837592870027159654528536, 9.529742981930146919351981428339, 10.72623539353959444592446458878, 11.61891136459377321833752625181, 13.04130189271651886094265435114, 13.86114257142942996092374959620

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