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Properties

Label 2-63-63.41-c3-0-4
Degree $2$
Conductor $63$
Sign $0.0663 - 0.997i$
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  + (−3.38 + 1.95i)2-s + (−4.35 − 2.83i)3-s + (3.63 − 6.29i)4-s + (5.82 − 10.0i)5-s + (20.2 + 1.07i)6-s + (−2.49 + 18.3i)7-s − 2.86i·8-s + (10.9 + 24.6i)9-s + 45.5i·10-s + (−41.2 + 23.8i)11-s + (−33.6 + 17.1i)12-s + (60.2 + 34.7i)13-s + (−27.3 − 66.9i)14-s + (−53.9 + 27.4i)15-s + (34.6 + 60.0i)16-s + 25.6·17-s + ⋯
L(s)  = 1  + (−1.19 + 0.690i)2-s + (−0.838 − 0.545i)3-s + (0.454 − 0.786i)4-s + (0.521 − 0.902i)5-s + (1.37 + 0.0729i)6-s + (−0.134 + 0.990i)7-s − 0.126i·8-s + (0.405 + 0.913i)9-s + 1.43i·10-s + (−1.13 + 0.653i)11-s + (−0.809 + 0.412i)12-s + (1.28 + 0.741i)13-s + (−0.523 − 1.27i)14-s + (−0.928 + 0.472i)15-s + (0.541 + 0.938i)16-s + 0.366·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0663 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0663 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.0663 - 0.997i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.0663 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.387198 + 0.362295i\)
\(L(\frac12)\) \(\approx\) \(0.387198 + 0.362295i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.35 + 2.83i)T \)
7 \( 1 + (2.49 - 18.3i)T \)
good2 \( 1 + (3.38 - 1.95i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-5.82 + 10.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (41.2 - 23.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-60.2 - 34.7i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 25.6T + 4.91e3T^{2} \)
19 \( 1 - 41.1iT - 6.85e3T^{2} \)
23 \( 1 + (-145. - 84.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-28.2 + 16.2i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (148. + 85.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 3.86T + 5.06e4T^{2} \)
41 \( 1 + (168. - 292. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-207. - 359. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (74.3 + 128. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 59.7iT - 1.48e5T^{2} \)
59 \( 1 + (282. - 488. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-571. + 330. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-145. + 252. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 3.05iT - 3.57e5T^{2} \)
73 \( 1 + 506. iT - 3.89e5T^{2} \)
79 \( 1 + (96.6 + 167. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-216. - 374. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 924.T + 7.04e5T^{2} \)
97 \( 1 + (943. - 544. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−15.30352409135830812041269961213, −13.23668401004023453082868628639, −12.65050933322240130996475542495, −11.20028255499908758279885480100, −9.791188731875043631790718980965, −8.806707032291379162723752944257, −7.69029206823748255931395870065, −6.28430135917670007220342713873, −5.23798096494532862196480887659, −1.44746906452964636735342156018, 0.66037830098073589816340190096, 3.18705654782984881123359185514, 5.49708717360265024692105318726, 7.02068835720095784254799570136, 8.625903203065400565935606359449, 10.08206876039095634198118437217, 10.76426221679798741893609191201, 11.01933488778448185505952703678, 12.88719487167302316786119087003, 14.17059847839025122198429260067

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