Properties

Degree $2$
Conductor $63$
Sign $-0.977 - 0.209i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.648 − 0.374i)2-s + (−3.71 − 6.44i)4-s + (−5.42 + 9.40i)5-s + (−18.2 − 3.24i)7-s + 11.5i·8-s + (7.04 − 4.06i)10-s + (−44.9 + 25.9i)11-s − 32.1i·13-s + (10.6 + 8.93i)14-s + (−25.4 + 44.0i)16-s + (−40.7 − 70.5i)17-s + (−0.0420 − 0.0242i)19-s + 80.7·20-s + 38.8·22-s + (77.3 + 44.6i)23-s + ⋯
L(s)  = 1  + (−0.229 − 0.132i)2-s + (−0.464 − 0.805i)4-s + (−0.485 + 0.840i)5-s + (−0.984 − 0.175i)7-s + 0.511i·8-s + (0.222 − 0.128i)10-s + (−1.23 + 0.710i)11-s − 0.686i·13-s + (0.202 + 0.170i)14-s + (−0.397 + 0.688i)16-s + (−0.581 − 1.00i)17-s + (−0.000507 − 0.000293i)19-s + 0.902·20-s + 0.376·22-s + (0.701 + 0.404i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.977 - 0.209i$
Motivic weight: \(3\)
Character: $\chi_{63} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.977 - 0.209i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00307351 + 0.0290393i\)
\(L(\frac12)\) \(\approx\) \(0.00307351 + 0.0290393i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.2 + 3.24i)T \)
good2 \( 1 + (0.648 + 0.374i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (5.42 - 9.40i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (44.9 - 25.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 32.1iT - 2.19e3T^{2} \)
17 \( 1 + (40.7 + 70.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (0.0420 + 0.0242i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-77.3 - 44.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 175. iT - 2.43e4T^{2} \)
31 \( 1 + (-186. + 107. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (32.2 - 55.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 411.T + 6.89e4T^{2} \)
43 \( 1 + 234.T + 7.95e4T^{2} \)
47 \( 1 + (316. - 547. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (230. - 132. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (175. + 304. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (673. + 389. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-98.0 - 169. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 142. iT - 3.57e5T^{2} \)
73 \( 1 + (-676. + 390. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (644. - 1.11e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 235.T + 5.71e5T^{2} \)
89 \( 1 + (-335. + 580. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 655. iT - 9.12e5T^{2} \)
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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

−13.79996186858026505816292990444, −12.95831936992108288459544277682, −11.32431343161928418992037746135, −10.29544557428622257505311578074, −9.543212703703829545921802605557, −7.82485630699828343069453982483, −6.51835466733230346963024814608, −4.92178020387228777197783631157, −2.89687491927329116345255401691, −0.02081761808774684410761913369, 3.30845585859995166577590503523, 4.84517377772907053893817720746, 6.73562025464620021803029694741, 8.312101158937948348142680579607, 8.869378616754395165646938156646, 10.37498885921724212399463412124, 12.00328496761675170443358484879, 12.88471711072122726124277960981, 13.53708395275627570881158962755, 15.40495589518923463557394529530

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