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Properties

Label 2-63-63.5-c3-0-4
Degree $2$
Conductor $63$
Sign $-0.660 - 0.751i$
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  + 1.83i·2-s + (−1.68 + 4.91i)3-s + 4.62·4-s + (0.207 + 0.359i)5-s + (−9.03 − 3.10i)6-s + (5.26 + 17.7i)7-s + 23.1i·8-s + (−21.2 − 16.5i)9-s + (−0.659 + 0.381i)10-s + (−42.9 − 24.8i)11-s + (−7.80 + 22.7i)12-s + (−1.43 − 0.828i)13-s + (−32.6 + 9.67i)14-s + (−2.11 + 0.412i)15-s − 5.67·16-s + (20.6 + 35.7i)17-s + ⋯
L(s)  = 1  + 0.649i·2-s + (−0.324 + 0.945i)3-s + 0.577·4-s + (0.0185 + 0.0321i)5-s + (−0.614 − 0.211i)6-s + (0.284 + 0.958i)7-s + 1.02i·8-s + (−0.788 − 0.614i)9-s + (−0.0208 + 0.0120i)10-s + (−1.17 − 0.679i)11-s + (−0.187 + 0.546i)12-s + (−0.0306 − 0.0176i)13-s + (−0.623 + 0.184i)14-s + (−0.0363 + 0.00709i)15-s − 0.0886·16-s + (0.294 + 0.510i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.751i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.660 - 0.751i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.660 - 0.751i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.588208 + 1.30019i\)
\(L(\frac12)\) \(\approx\) \(0.588208 + 1.30019i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.68 - 4.91i)T \)
7 \( 1 + (-5.26 - 17.7i)T \)
good2 \( 1 - 1.83iT - 8T^{2} \)
5 \( 1 + (-0.207 - 0.359i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (42.9 + 24.8i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (1.43 + 0.828i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-20.6 - 35.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-130. - 75.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-114. + 65.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (136. - 78.8i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 18.4iT - 2.97e4T^{2} \)
37 \( 1 + (-173. + 301. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-16.9 + 29.4i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-29.5 - 51.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 216.T + 1.03e5T^{2} \)
53 \( 1 + (174. - 100. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 299.T + 2.05e5T^{2} \)
61 \( 1 + 80.4iT - 2.26e5T^{2} \)
67 \( 1 + 257.T + 3.00e5T^{2} \)
71 \( 1 + 1.03e3iT - 3.57e5T^{2} \)
73 \( 1 + (711. - 410. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 105.T + 4.93e5T^{2} \)
83 \( 1 + (245. + 424. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-754. + 1.30e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.08e3 - 627. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−15.05485773124242187760163782596, −14.31783018515004644535639909587, −12.44470588197695757302452222524, −11.32227453362106578547686916225, −10.45729591862204159734508025400, −8.917574604724271178363878618939, −7.77003570949241226589024423730, −5.94065032427813024202298509384, −5.24142768585185776583884670556, −2.92298354219619560129285503436, 1.10725017794928905327598619209, 2.86151759657438361517862142767, 5.20882697637244481515602910765, 7.10148799795711313969058555307, 7.57488154678744586545537324403, 9.751082124840604464410102101902, 10.99763532921327348584512157020, 11.64880066272952128733461837599, 12.96522534769418613498253528361, 13.54771706357100820811308172186

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