Properties

Degree $2$
Conductor $63$
Sign $0.701 + 0.712i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (2.00 + 3.46i)4-s + (3.5 − 6.06i)5-s + (14 − 12.1i)7-s + 24·8-s + (−7 − 12.1i)10-s + (−2.5 − 4.33i)11-s − 14·13-s + (−7 − 36.3i)14-s + (8.00 − 13.8i)16-s + (−10.5 − 18.1i)17-s + (−24.5 + 42.4i)19-s + 28·20-s − 10·22-s + (−79.5 + 137. i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.250 + 0.433i)4-s + (0.313 − 0.542i)5-s + (0.755 − 0.654i)7-s + 1.06·8-s + (−0.221 − 0.383i)10-s + (−0.0685 − 0.118i)11-s − 0.298·13-s + (−0.133 − 0.694i)14-s + (0.125 − 0.216i)16-s + (−0.149 − 0.259i)17-s + (−0.295 + 0.512i)19-s + 0.313·20-s − 0.0969·22-s + (−0.720 + 1.24i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.701 + 0.712i$
Motivic weight: \(3\)
Character: $\chi_{63} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.84467 - 0.773070i\)
\(L(\frac12)\) \(\approx\) \(1.84467 - 0.773070i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14 + 12.1i)T \)
good2 \( 1 + (-1 + 1.73i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-3.5 + 6.06i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 14T + 2.19e3T^{2} \)
17 \( 1 + (10.5 + 18.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (24.5 - 42.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (79.5 - 137. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 58T + 2.43e4T^{2} \)
31 \( 1 + (73.5 + 127. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (109.5 - 189. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 350T + 6.89e4T^{2} \)
43 \( 1 + 124T + 7.95e4T^{2} \)
47 \( 1 + (-262.5 + 454. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-151.5 - 262. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (52.5 + 90.9i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-206.5 + 357. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (207.5 + 359. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 432T + 3.57e5T^{2} \)
73 \( 1 + (-556.5 - 963. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-51.5 + 89.2i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.09e3T + 5.71e5T^{2} \)
89 \( 1 + (164.5 - 284. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 882T + 9.12e5T^{2} \)
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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

−13.92538997931754736536572744171, −13.21492657460429433862257146774, −12.03613992434757571030065168987, −11.15394390449886121614054168901, −9.959124515911440903195033045197, −8.327961980626200401606484670613, −7.21741002012368570943842159628, −5.17525991275837360078382387211, −3.77791423809809465049800301767, −1.75382809136259760563347454696, 2.17132250947332270388151761296, 4.73313483146484211874996179363, 5.98008470630566320886347902001, 7.11026931262591983094923791019, 8.558030329400515127888650673747, 10.19612924594517746674523002482, 11.09246777481280212376019510064, 12.46219209517447284072293189274, 13.95583680330313333603492418147, 14.64814336432912702734701116954

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