Properties

Degree $2$
Conductor $63$
Sign $0.699 - 0.714i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.799 + 1.38i)2-s + (2.72 + 4.71i)4-s + (9.14 − 15.8i)5-s + (12.3 + 13.7i)7-s − 21.4·8-s + (14.6 + 25.3i)10-s + (30.6 + 53.0i)11-s + 32.4·13-s + (−28.9 + 6.15i)14-s + (−4.61 + 7.99i)16-s + (−40.6 − 70.4i)17-s + (10.4 − 18.1i)19-s + 99.6·20-s − 97.9·22-s + (−16.8 + 29.2i)23-s + ⋯
L(s)  = 1  + (−0.282 + 0.489i)2-s + (0.340 + 0.589i)4-s + (0.818 − 1.41i)5-s + (0.669 + 0.743i)7-s − 0.949·8-s + (0.462 + 0.800i)10-s + (0.839 + 1.45i)11-s + 0.692·13-s + (−0.552 + 0.117i)14-s + (−0.0721 + 0.124i)16-s + (−0.580 − 1.00i)17-s + (0.126 − 0.218i)19-s + 1.11·20-s − 0.948·22-s + (−0.152 + 0.264i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.699 - 0.714i$
Motivic weight: \(3\)
Character: $\chi_{63} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.699 - 0.714i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.44581 + 0.608225i\)
\(L(\frac12)\) \(\approx\) \(1.44581 + 0.608225i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-12.3 - 13.7i)T \)
good2 \( 1 + (0.799 - 1.38i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-9.14 + 15.8i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-30.6 - 53.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 32.4T + 2.19e3T^{2} \)
17 \( 1 + (40.6 + 70.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-10.4 + 18.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (16.8 - 29.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 52.0T + 2.43e4T^{2} \)
31 \( 1 + (96.9 + 167. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-133. + 231. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 203.T + 6.89e4T^{2} \)
43 \( 1 + 21.9T + 7.95e4T^{2} \)
47 \( 1 + (123. - 214. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-70.4 - 121. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (110. + 191. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (326. - 565. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (302. + 523. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 716.T + 3.57e5T^{2} \)
73 \( 1 + (194. + 336. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-144. + 250. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 115.T + 5.71e5T^{2} \)
89 \( 1 + (-469. + 813. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 120.T + 9.12e5T^{2} \)
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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

−14.83904175885881651841784752384, −13.34642877200258760841220044649, −12.36733958475721945634189472749, −11.55403279033007760146491274163, −9.360190205889010047133058047694, −8.914841705782565597332987985972, −7.53026429829932286598740114069, −5.98582912559248432474082167657, −4.59905531405162958262248200220, −1.92738238983978013877255190590, 1.57294970242384895169638812720, 3.38280915997644845325388825441, 5.94670761193068942015686945801, 6.73128380195389277674391454048, 8.639285277510095705888101378861, 10.14269779390654404572196722240, 10.87057915118768391885426027989, 11.45964343739233915868801076788, 13.56434087493483986490197843542, 14.30440120290466387541868173874

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