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Properties

Label 2-63-63.4-c3-0-17
Degree $2$
Conductor $63$
Sign $0.0477 + 0.998i$
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  + (2.53 − 4.39i)2-s + (1.11 + 5.07i)3-s + (−8.90 − 15.4i)4-s + 18.4·5-s + (25.1 + 7.97i)6-s + (−9.68 − 15.7i)7-s − 49.8·8-s + (−24.5 + 11.3i)9-s + (46.9 − 81.2i)10-s + 17.1·11-s + (68.3 − 62.4i)12-s + (−27.2 + 47.2i)13-s + (−94.0 + 2.52i)14-s + (20.6 + 93.7i)15-s + (−55.3 + 95.7i)16-s + (−7.61 + 13.1i)17-s + ⋯
L(s)  = 1  + (0.898 − 1.55i)2-s + (0.214 + 0.976i)3-s + (−1.11 − 1.92i)4-s + 1.65·5-s + (1.71 + 0.542i)6-s + (−0.523 − 0.852i)7-s − 2.20·8-s + (−0.907 + 0.419i)9-s + (1.48 − 2.56i)10-s + 0.470·11-s + (1.64 − 1.50i)12-s + (−0.582 + 1.00i)13-s + (−1.79 + 0.0482i)14-s + (0.355 + 1.61i)15-s + (−0.864 + 1.49i)16-s + (−0.108 + 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0477 + 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0477 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.0477 + 0.998i$
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: \((2,\ 63,\ (\ :3/2),\ 0.0477 + 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.80966 - 1.72530i\)
\(L(\frac12)\) \(\approx\) \(1.80966 - 1.72530i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.11 - 5.07i)T \)
7 \( 1 + (9.68 + 15.7i)T \)
good2 \( 1 + (-2.53 + 4.39i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 18.4T + 125T^{2} \)
11 \( 1 - 17.1T + 1.33e3T^{2} \)
13 \( 1 + (27.2 - 47.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (7.61 - 13.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-43.7 - 75.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 96.6T + 1.21e4T^{2} \)
29 \( 1 + (-57.3 - 99.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (25.0 + 43.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-3.58 - 6.20i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (87.4 - 151. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (77.4 + 134. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-58.8 + 101. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (1.27 - 2.20i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (206. + 356. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-355. + 616. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (18.8 + 32.6i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 290.T + 3.57e5T^{2} \)
73 \( 1 + (-209. + 363. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-537. + 931. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (634. + 1.09e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-5.74 - 9.95i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-935. - 1.61e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.10124644957028711208871844575, −13.23910829699413699541024481331, −11.91712236456503992787459698816, −10.58144682807302825297610440113, −9.907973833541522289947966322909, −9.307596896475943061444015784207, −6.15130430908272196971373754457, −4.82057442506652467779863415644, −3.52773133445510913819158413726, −1.92886850849527777570983135190, 2.70975930924888520314868786782, 5.39109317055792046353950186869, 6.08301979458835424358790782208, 7.05862912213780434279772326526, 8.516810348397673059441637327455, 9.606838612181678293076416027924, 12.15898081283693191857262110742, 13.04904887038010850788117042186, 13.71164052850611514591152327680, 14.53424641898616880756891128980

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