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Properties

Label 2-63-7.2-c5-0-12
Degree $2$
Conductor $63$
Sign $-0.440 + 0.897i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.54 − 6.13i)2-s + (−9.12 − 15.8i)4-s + (41.1 − 71.1i)5-s + (112. + 64.3i)7-s + 97.4·8-s + (−291. − 504. i)10-s + (−176. − 305. i)11-s − 885.·13-s + (793. − 463. i)14-s + (637. − 1.10e3i)16-s + (212. + 368. i)17-s + (781. − 1.35e3i)19-s − 1.50e3·20-s − 2.49e3·22-s + (−1.39e3 + 2.41e3i)23-s + ⋯
L(s)  = 1  + (0.626 − 1.08i)2-s + (−0.285 − 0.494i)4-s + (0.735 − 1.27i)5-s + (0.868 + 0.496i)7-s + 0.538·8-s + (−0.921 − 1.59i)10-s + (−0.438 − 0.760i)11-s − 1.45·13-s + (1.08 − 0.631i)14-s + (0.622 − 1.07i)16-s + (0.178 + 0.308i)17-s + (0.496 − 0.859i)19-s − 0.839·20-s − 1.09·22-s + (−0.549 + 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.440 + 0.897i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.440 + 0.897i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.50864 - 2.41989i\)
\(L(\frac12)\) \(\approx\) \(1.50864 - 2.41989i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-112. - 64.3i)T \)
good2 \( 1 + (-3.54 + 6.13i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (-41.1 + 71.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (176. + 305. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 885.T + 3.71e5T^{2} \)
17 \( 1 + (-212. - 368. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-781. + 1.35e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.39e3 - 2.41e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 3.67e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.79e3 - 3.11e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (7.14e3 - 1.23e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.43e4T + 1.15e8T^{2} \)
43 \( 1 - 7.58e3T + 1.47e8T^{2} \)
47 \( 1 + (-2.88e3 + 4.99e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.26e4 - 2.19e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (2.16e4 + 3.74e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (9.72e3 - 1.68e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.47e4 - 2.54e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 5.16e4T + 1.80e9T^{2} \)
73 \( 1 + (1.89e4 + 3.28e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (2.68e4 - 4.65e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 8.59e4T + 3.93e9T^{2} \)
89 \( 1 + (-1.04e4 + 1.80e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 9.75e4T + 8.58e9T^{2} \)
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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

−13.41736447353629374638539933491, −12.36674004211710286802592104189, −11.71040228880556361952111852150, −10.35343249271205582140486141614, −9.140339907189686750800232386838, −7.83728107094864646617134767727, −5.41657830004984424859171779067, −4.69273931557310316446032092662, −2.62174599366163637751140324053, −1.25811851216410797731156737387, 2.25771672610148598719412816747, 4.51191873812479901716679137682, 5.75784843746866856448144743560, 7.08382602622533976583093300851, 7.70373413125957173246587058383, 9.977601256732319963654106604356, 10.67938760098802880160472721811, 12.33741132667364789372114958257, 13.88956751073263425312157135560, 14.41004958346859163226376460456

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