Properties

Degree $2$
Conductor $63$
Sign $-0.991 - 0.126i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 2.59i)2-s + (−0.5 − 0.866i)4-s + (−1.5 + 2.59i)5-s + (−3.5 + 18.1i)7-s − 21·8-s + (−4.5 − 7.79i)10-s + (−7.5 − 12.9i)11-s − 64·13-s + (−42 − 36.3i)14-s + (35.5 − 61.4i)16-s + (42 + 72.7i)17-s + (8 − 13.8i)19-s + 3.00·20-s + 45·22-s + (−42 + 72.7i)23-s + ⋯
L(s)  = 1  + (−0.530 + 0.918i)2-s + (−0.0625 − 0.108i)4-s + (−0.134 + 0.232i)5-s + (−0.188 + 0.981i)7-s − 0.928·8-s + (−0.142 − 0.246i)10-s + (−0.205 − 0.356i)11-s − 1.36·13-s + (−0.801 − 0.694i)14-s + (0.554 − 0.960i)16-s + (0.599 + 1.03i)17-s + (0.0965 − 0.167i)19-s + 0.0335·20-s + 0.436·22-s + (−0.380 + 0.659i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.991 - 0.126i$
Motivic weight: \(3\)
Character: $\chi_{63} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0495088 + 0.780156i\)
\(L(\frac12)\) \(\approx\) \(0.0495088 + 0.780156i\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.5 - 18.1i)T \)
good2 \( 1 + (1.5 - 2.59i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (7.5 + 12.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 64T + 2.19e3T^{2} \)
17 \( 1 + (-42 - 72.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-8 + 13.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (42 - 72.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 297T + 2.43e4T^{2} \)
31 \( 1 + (-126.5 - 219. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-158 + 273. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 360T + 6.89e4T^{2} \)
43 \( 1 - 26T + 7.95e4T^{2} \)
47 \( 1 + (15 - 25.9i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-181.5 - 314. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-59 + 102. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-185 - 320. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 342T + 3.57e5T^{2} \)
73 \( 1 + (181 + 313. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (233.5 - 404. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 477T + 5.71e5T^{2} \)
89 \( 1 + (-453 + 784. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 503T + 9.12e5T^{2} \)
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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

−15.23103664677289981250143323834, −14.34520857597417977910322597573, −12.57850558488342570499673910701, −11.83379622712066472602735915478, −10.12939170901032552824418014413, −8.893526795560829037851986070013, −7.88830335050283914168109251131, −6.65971297350629044580055596238, −5.39935078514453368454433006921, −2.93789211719097433935532809527, 0.62473778199903321829856653931, 2.71962746093523372546122913870, 4.70459420574296370063987558869, 6.71351177861983897343153583355, 8.147107920806730178073587235006, 9.857748596758354895774400295598, 10.16448849365863446054792451355, 11.67947089290166356129346565283, 12.41663850427594300802213263994, 13.85626431182820684584894852303

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