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Properties

Label 2-63-63.20-c3-0-21
Degree $2$
Conductor $63$
Sign $-0.459 - 0.888i$
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  + (−2.31 − 1.33i)2-s + (−3.10 − 4.16i)3-s + (−0.417 − 0.723i)4-s + (−0.223 − 0.386i)5-s + (1.62 + 13.8i)6-s + (−7.50 − 16.9i)7-s + 23.6i·8-s + (−7.70 + 25.8i)9-s + 1.19i·10-s + (34.2 + 19.7i)11-s + (−1.71 + 3.98i)12-s + (−68.4 + 39.5i)13-s + (−5.25 + 49.2i)14-s + (−0.917 + 2.13i)15-s + (28.3 − 49.0i)16-s − 9.74·17-s + ⋯
L(s)  = 1  + (−0.819 − 0.473i)2-s + (−0.597 − 0.801i)3-s + (−0.0522 − 0.0904i)4-s + (−0.0199 − 0.0345i)5-s + (0.110 + 0.939i)6-s + (−0.405 − 0.914i)7-s + 1.04i·8-s + (−0.285 + 0.958i)9-s + 0.0377i·10-s + (0.939 + 0.542i)11-s + (−0.0412 + 0.0959i)12-s + (−1.46 + 0.843i)13-s + (−0.100 + 0.941i)14-s + (−0.0157 + 0.0366i)15-s + (0.442 − 0.766i)16-s − 0.139·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.459 - 0.888i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.459 - 0.888i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0692930 + 0.113829i\)
\(L(\frac12)\) \(\approx\) \(0.0692930 + 0.113829i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.10 + 4.16i)T \)
7 \( 1 + (7.50 + 16.9i)T \)
good2 \( 1 + (2.31 + 1.33i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (0.223 + 0.386i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-34.2 - 19.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (68.4 - 39.5i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 9.74T + 4.91e3T^{2} \)
19 \( 1 + 73.1iT - 6.85e3T^{2} \)
23 \( 1 + (126. - 73.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (134. + 77.4i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (9.87 - 5.70i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + (53.3 + 92.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (45.6 - 78.9i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (276. - 479. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 239. iT - 1.48e5T^{2} \)
59 \( 1 + (126. + 218. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-342. - 197. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-13.5 - 23.4i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 348. iT - 3.57e5T^{2} \)
73 \( 1 + 923. iT - 3.89e5T^{2} \)
79 \( 1 + (-280. + 485. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-281. + 487. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 644.T + 7.04e5T^{2} \)
97 \( 1 + (427. + 246. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.69688962196948508913168212652, −12.27366443944525239679081796221, −11.42539633760636915433724320777, −10.21154072677275927116268628787, −9.279225120526381828225768624630, −7.62368314887841779714829589552, −6.57742958949392083010760917207, −4.73293145741771925369817004998, −1.87493513639137911152867723617, −0.12543352454029934610053175981, 3.57810978068381850389353059187, 5.45166050613679395599685257427, 6.78741382569081379267060920917, 8.426900978107943100559362493647, 9.422071427631433942056476966326, 10.25989216139996901844174791530, 11.88545650715688623329649750751, 12.63374135934512238548133766466, 14.58703032803169089485761260220, 15.46353278110526025794741344105

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