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Properties

Label 2-63-63.38-c3-0-2
Degree $2$
Conductor $63$
Sign $-0.224 + 0.974i$
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  + 5.07i·2-s + (−4.42 + 2.72i)3-s − 17.7·4-s + (4.21 − 7.29i)5-s + (−13.8 − 22.4i)6-s + (−4.29 + 18.0i)7-s − 49.6i·8-s + (12.1 − 24.0i)9-s + (37.0 + 21.3i)10-s + (−39.7 + 22.9i)11-s + (78.6 − 48.3i)12-s + (18.3 − 10.6i)13-s + (−91.4 − 21.8i)14-s + (1.20 + 43.7i)15-s + 109.·16-s + (−8.26 + 14.3i)17-s + ⋯
L(s)  = 1  + 1.79i·2-s + (−0.851 + 0.523i)3-s − 2.22·4-s + (0.376 − 0.652i)5-s + (−0.940 − 1.52i)6-s + (−0.232 + 0.972i)7-s − 2.19i·8-s + (0.451 − 0.892i)9-s + (1.17 + 0.676i)10-s + (−1.08 + 0.628i)11-s + (1.89 − 1.16i)12-s + (0.392 − 0.226i)13-s + (−1.74 − 0.416i)14-s + (0.0208 + 0.753i)15-s + 1.71·16-s + (−0.117 + 0.204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.224 + 0.974i$
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: \((2,\ 63,\ (\ :3/2),\ -0.224 + 0.974i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.333636 - 0.419389i\)
\(L(\frac12)\) \(\approx\) \(0.333636 - 0.419389i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.42 - 2.72i)T \)
7 \( 1 + (4.29 - 18.0i)T \)
good2 \( 1 - 5.07iT - 8T^{2} \)
5 \( 1 + (-4.21 + 7.29i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (39.7 - 22.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-18.3 + 10.6i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (8.26 - 14.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (49.7 - 28.7i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (167. + 96.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (47.1 + 27.2i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 294. iT - 2.97e4T^{2} \)
37 \( 1 + (-185. - 320. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-166. - 287. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-104. + 180. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 419.T + 1.03e5T^{2} \)
53 \( 1 + (275. + 159. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 298.T + 2.05e5T^{2} \)
61 \( 1 - 226. iT - 2.26e5T^{2} \)
67 \( 1 + 99.7T + 3.00e5T^{2} \)
71 \( 1 - 176. iT - 3.57e5T^{2} \)
73 \( 1 + (-142. - 82.2i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 374.T + 4.93e5T^{2} \)
83 \( 1 + (-457. + 792. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-67.8 - 117. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (377. + 217. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−15.58794760120339084016829823086, −14.67046323373308465886204987887, −13.12552303078679829123215441715, −12.33042422949851013376145065962, −10.31155320950662347530518608277, −9.173359006764751074368160694380, −8.111020345771540517777436452820, −6.41790592480155081875288040943, −5.57124986013622512183044222257, −4.61716650501588499961671011419, 0.40303031120601101875383979088, 2.31371581950224952058291132026, 4.12096090691038338171413377260, 5.93204032193802437158034763344, 7.71856411323817941206383915650, 9.668813310924718845806823979503, 10.81146842182560707390366490517, 11.00882452966917351102232877665, 12.43284052923266360965682531379, 13.41090021572883348030201924077

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