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Properties

Label 2-63-63.38-c3-0-15
Degree $2$
Conductor $63$
Sign $-0.687 + 0.726i$
Analytic cond. $3.71712$
Root an. cond. $1.92798$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.46i·2-s + (−2.57 + 4.51i)3-s − 4.01·4-s + (9.02 − 15.6i)5-s + (15.6 + 8.92i)6-s + (−18.1 − 3.61i)7-s − 13.8i·8-s + (−13.7 − 23.2i)9-s + (−54.1 − 31.2i)10-s + (−10.2 + 5.89i)11-s + (10.3 − 18.1i)12-s + (45.9 − 26.5i)13-s + (−12.5 + 62.9i)14-s + (47.3 + 81.0i)15-s − 79.9·16-s + (5.93 − 10.2i)17-s + ⋯
L(s)  = 1  − 1.22i·2-s + (−0.495 + 0.868i)3-s − 0.501·4-s + (0.807 − 1.39i)5-s + (1.06 + 0.607i)6-s + (−0.980 − 0.194i)7-s − 0.610i·8-s + (−0.508 − 0.861i)9-s + (−1.71 − 0.989i)10-s + (−0.279 + 0.161i)11-s + (0.248 − 0.435i)12-s + (0.980 − 0.566i)13-s + (−0.238 + 1.20i)14-s + (0.814 + 1.39i)15-s − 1.24·16-s + (0.0847 − 0.146i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.687 + 0.726i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.687 + 0.726i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.484582 - 1.12570i\)
\(L(\frac12)\) \(\approx\) \(0.484582 - 1.12570i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.57 - 4.51i)T \)
7 \( 1 + (18.1 + 3.61i)T \)
good2 \( 1 + 3.46iT - 8T^{2} \)
5 \( 1 + (-9.02 + 15.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (10.2 - 5.89i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-45.9 + 26.5i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-5.93 + 10.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (12.1 - 7.04i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-139. - 80.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-131. - 75.8i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 90.1iT - 2.97e4T^{2} \)
37 \( 1 + (-185. - 320. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (129. + 223. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (173. - 300. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 71.6T + 1.03e5T^{2} \)
53 \( 1 + (-143. - 82.9i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 575.T + 2.05e5T^{2} \)
61 \( 1 + 496. iT - 2.26e5T^{2} \)
67 \( 1 + 700.T + 3.00e5T^{2} \)
71 \( 1 + 1.13e3iT - 3.57e5T^{2} \)
73 \( 1 + (-281. - 162. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 523.T + 4.93e5T^{2} \)
83 \( 1 + (-410. + 710. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (181. + 313. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-273. - 157. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.37221697908095073606681952538, −12.88027468238131326136500827704, −11.75376371221478500858782569871, −10.50225561798498120978950256297, −9.725278302293086058541650690606, −8.878891231388694003232093795960, −6.23148388545293110080371430401, −4.85020027608109679541867012826, −3.29939774542057414647634240248, −0.937213126120314867887425190830, 2.60227266530473184519987816977, 5.70931560709078706733341449104, 6.52124009240936485926195890575, 7.06005901174877989928331448623, 8.667974635030077580211123788649, 10.41362588631316687443048985906, 11.41697045110180942286971657088, 13.08978211811395275688198907618, 13.89215149091898587584209780276, 14.84459843647351391944301898769

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