Properties

Degree $2$
Conductor $63$
Sign $-0.903 + 0.428i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.47 − 4.28i)2-s + (−2.38 + 4.61i)3-s + (−8.26 − 14.3i)4-s + (−8.06 − 13.9i)5-s + (13.8 + 21.6i)6-s + (3.5 − 6.06i)7-s − 42.1·8-s + (−15.6 − 22.0i)9-s − 79.9·10-s + (15.3 − 26.5i)11-s + (85.7 − 4.01i)12-s + (39.0 + 67.5i)13-s + (−17.3 − 30.0i)14-s + (83.7 − 3.92i)15-s + (−38.3 + 66.4i)16-s + 106.·17-s + ⋯
L(s)  = 1  + (0.875 − 1.51i)2-s + (−0.458 + 0.888i)3-s + (−1.03 − 1.78i)4-s + (−0.721 − 1.25i)5-s + (0.945 + 1.47i)6-s + (0.188 − 0.327i)7-s − 1.86·8-s + (−0.578 − 0.815i)9-s − 2.52·10-s + (0.419 − 0.727i)11-s + (2.06 − 0.0966i)12-s + (0.832 + 1.44i)13-s + (−0.330 − 0.573i)14-s + (1.44 − 0.0675i)15-s + (−0.599 + 1.03i)16-s + 1.52·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.903 + 0.428i$
Motivic weight: \(3\)
Character: $\chi_{63} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.903 + 0.428i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.345044 - 1.53329i\)
\(L(\frac12)\) \(\approx\) \(0.345044 - 1.53329i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.38 - 4.61i)T \)
7 \( 1 + (-3.5 + 6.06i)T \)
good2 \( 1 + (-2.47 + 4.28i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (8.06 + 13.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-15.3 + 26.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-39.0 - 67.5i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 106.T + 4.91e3T^{2} \)
19 \( 1 + 49.5T + 6.85e3T^{2} \)
23 \( 1 + (19.5 + 33.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (2.87 - 4.97i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (92.2 + 159. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 91.0T + 5.06e4T^{2} \)
41 \( 1 + (-41.4 - 71.7i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-45.6 + 79.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (118. - 206. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 497.T + 1.48e5T^{2} \)
59 \( 1 + (-2.10 - 3.64i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (313. - 542. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-339. - 587. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 747.T + 3.57e5T^{2} \)
73 \( 1 + 23.2T + 3.89e5T^{2} \)
79 \( 1 + (77.0 - 133. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (141. - 244. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 111.T + 7.04e5T^{2} \)
97 \( 1 + (555. - 961. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.77718268958199817649516610269, −12.49228102737460180703576297075, −11.68746712549630270413108448141, −11.03561629773945933673398352199, −9.687538083977717084804638612188, −8.630537794036643976624870683009, −5.74036150707546940616554500275, −4.40140543946567529484870286678, −3.77458224081294579482075298037, −0.977155801124939165433842024963, 3.43612368671424180778522189786, 5.40659694524755187635941238677, 6.45955251174879542917671390602, 7.46579151438800971572891788871, 8.159276056072019989845949104347, 10.68223530975118138098603098310, 12.04336772556858680767354560938, 12.89772603463521842740459808438, 14.17467588313620275745093188975, 14.88227187182055378938122164078

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