Properties

Label 2-63-63.47-c3-0-20
Degree $2$
Conductor $63$
Sign $-0.998 + 0.0487i$
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  + (0.725 − 0.418i)2-s + (0.141 − 5.19i)3-s + (−3.64 + 6.32i)4-s − 21.9·5-s + (−2.07 − 3.82i)6-s + (2.19 − 18.3i)7-s + 12.8i·8-s + (−26.9 − 1.47i)9-s + (−15.8 + 9.17i)10-s + 15.4i·11-s + (32.3 + 19.8i)12-s + (33.3 − 19.2i)13-s + (−6.10 − 14.2i)14-s + (−3.11 + 113. i)15-s + (−23.8 − 41.2i)16-s + (−34.8 − 60.3i)17-s + ⋯
L(s)  = 1  + (0.256 − 0.148i)2-s + (0.0273 − 0.999i)3-s + (−0.456 + 0.790i)4-s − 1.96·5-s + (−0.140 − 0.260i)6-s + (0.118 − 0.992i)7-s + 0.566i·8-s + (−0.998 − 0.0546i)9-s + (−0.502 + 0.290i)10-s + 0.423i·11-s + (0.777 + 0.477i)12-s + (0.711 − 0.410i)13-s + (−0.116 − 0.272i)14-s + (−0.0535 + 1.95i)15-s + (−0.372 − 0.644i)16-s + (−0.497 − 0.861i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0487i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.998 + 0.0487i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.998 + 0.0487i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00967326 - 0.396472i\)
\(L(\frac12)\) \(\approx\) \(0.00967326 - 0.396472i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.141 + 5.19i)T \)
7 \( 1 + (-2.19 + 18.3i)T \)
good2 \( 1 + (-0.725 + 0.418i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + 21.9T + 125T^{2} \)
11 \( 1 - 15.4iT - 1.33e3T^{2} \)
13 \( 1 + (-33.3 + 19.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (34.8 + 60.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (55.4 + 32.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 69.8iT - 1.21e4T^{2} \)
29 \( 1 + (168. + 97.3i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-68.0 - 39.3i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-3.34 + 5.79i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-9.21 - 15.9i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (12.2 - 21.1i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (138. + 239. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (95.3 - 55.0i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-176. + 306. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (460. - 265. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (262. - 453. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 43.3iT - 3.57e5T^{2} \)
73 \( 1 + (-54.9 + 31.7i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (606. + 1.04e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-111. + 192. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (35.2 - 61.0i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-483. - 279. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.57664630400805316353569615737, −12.82456988431336071954693307403, −11.73969153458672930752861052265, −11.13194989707199642777017315327, −8.677756512196376011483199905125, −7.74769205238622655999778990975, −7.10702460212586639170749025527, −4.50470207236335693832504432495, −3.34848246379791961869714838244, −0.25289017682461560032186310498, 3.69163205578123835035540787533, 4.66689912239422278271551548116, 6.16687247665802945590671466146, 8.319403399522640010523534353740, 8.984284274040321925761043638225, 10.70823549344725948351435329932, 11.40902157693033042119708917195, 12.68582622263067061684123625207, 14.43473445954065194076148921287, 15.15773537961358780768528252045

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