Properties

Degree $2$
Conductor $63$
Sign $0.440 - 0.897i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.91 + 2.26i)2-s + (6.22 + 10.7i)4-s + (0.632 − 1.09i)5-s + (13.4 + 12.7i)7-s + 20.1i·8-s + (4.95 − 2.86i)10-s + (−36.0 + 20.7i)11-s − 85.7i·13-s + (23.5 + 80.3i)14-s + (4.27 − 7.39i)16-s + (−38.8 − 67.3i)17-s + (42.1 + 24.3i)19-s + 15.7·20-s − 188.·22-s + (−78.7 − 45.4i)23-s + ⋯
L(s)  = 1  + (1.38 + 0.799i)2-s + (0.778 + 1.34i)4-s + (0.0566 − 0.0980i)5-s + (0.723 + 0.690i)7-s + 0.890i·8-s + (0.156 − 0.0905i)10-s + (−0.987 + 0.570i)11-s − 1.82i·13-s + (0.450 + 1.53i)14-s + (0.0667 − 0.115i)16-s + (−0.554 − 0.961i)17-s + (0.509 + 0.293i)19-s + 0.176·20-s − 1.82·22-s + (−0.714 − 0.412i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.440 - 0.897i$
Motivic weight: \(3\)
Character: $\chi_{63} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.440 - 0.897i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.41109 + 1.50178i\)
\(L(\frac12)\) \(\approx\) \(2.41109 + 1.50178i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-13.4 - 12.7i)T \)
good2 \( 1 + (-3.91 - 2.26i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-0.632 + 1.09i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (36.0 - 20.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 85.7iT - 2.19e3T^{2} \)
17 \( 1 + (38.8 + 67.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-42.1 - 24.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (78.7 + 45.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 151. iT - 2.43e4T^{2} \)
31 \( 1 + (-76.3 + 44.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (45.2 - 78.4i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 383.T + 6.89e4T^{2} \)
43 \( 1 + 227.T + 7.95e4T^{2} \)
47 \( 1 + (69.5 - 120. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-289. + 167. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-440. - 762. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-11.3 - 6.57i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-221. - 383. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 341. iT - 3.57e5T^{2} \)
73 \( 1 + (798. - 460. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-206. + 357. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 954.T + 5.71e5T^{2} \)
89 \( 1 + (-14.8 + 25.7i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.19e3iT - 9.12e5T^{2} \)
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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

−14.86528081425520072054618023150, −13.56650977830609243852289237642, −12.77094855120618244882187546580, −11.77944754807879496936629528816, −10.21929442158071428553723276599, −8.306182140216115328726092045412, −7.23596786134093464268992259311, −5.54572497277747491037255090519, −4.93483721452151411795379640969, −2.93193098160778545824400468752, 2.02458764618207795642870856539, 3.88228458493218805690499534006, 4.97914621771386468202520680003, 6.53235732880539887990214272141, 8.301035277844880099143398737473, 10.23082031743998147456442682615, 11.20720295957337356390467468726, 11.98209746379338236939006964464, 13.45593682508932429090181726056, 13.85295961555259030330739744112

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