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Properties

Label 2-63-9.5-c2-0-0
Degree $2$
Conductor $63$
Sign $-0.994 + 0.100i$
Analytic cond. $1.71662$
Root an. cond. $1.31020$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 + 1.00i)2-s + (−0.111 + 2.99i)3-s + (0.00971 − 0.0168i)4-s + (−4.27 − 2.46i)5-s + (−2.81 − 5.31i)6-s + (1.32 + 2.29i)7-s − 7.98i·8-s + (−8.97 − 0.665i)9-s + 9.90·10-s + (−9.29 + 5.36i)11-s + (0.0493 + 0.0309i)12-s + (−4.58 + 7.93i)13-s + (−4.59 − 2.65i)14-s + (7.87 − 12.5i)15-s + (8.03 + 13.9i)16-s + 18.4i·17-s + ⋯
L(s)  = 1  + (−0.868 + 0.501i)2-s + (−0.0370 + 0.999i)3-s + (0.00242 − 0.00420i)4-s + (−0.855 − 0.493i)5-s + (−0.468 − 0.886i)6-s + (0.188 + 0.327i)7-s − 0.997i·8-s + (−0.997 − 0.0739i)9-s + 0.990·10-s + (−0.845 + 0.488i)11-s + (0.00411 + 0.00258i)12-s + (−0.352 + 0.610i)13-s + (−0.328 − 0.189i)14-s + (0.525 − 0.836i)15-s + (0.502 + 0.870i)16-s + 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ -0.994 + 0.100i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0200428 - 0.398642i\)
\(L(\frac12)\) \(\approx\) \(0.0200428 - 0.398642i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.111 - 2.99i)T \)
7 \( 1 + (-1.32 - 2.29i)T \)
good2 \( 1 + (1.73 - 1.00i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (4.27 + 2.46i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (9.29 - 5.36i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (4.58 - 7.93i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 18.4iT - 289T^{2} \)
19 \( 1 - 16.4T + 361T^{2} \)
23 \( 1 + (-22.3 - 12.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (30.5 - 17.6i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (22.8 - 39.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 9.98T + 1.36e3T^{2} \)
41 \( 1 + (-39.7 - 22.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (39.1 + 67.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-25.6 + 14.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 45.5iT - 2.80e3T^{2} \)
59 \( 1 + (-17.2 - 9.94i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-53.3 - 92.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-36.3 + 63.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 79.1iT - 5.04e3T^{2} \)
73 \( 1 - 26.7T + 5.32e3T^{2} \)
79 \( 1 + (55.5 + 96.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-3.92 + 2.26i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 24.2iT - 7.92e3T^{2} \)
97 \( 1 + (-33.1 - 57.3i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−15.62376254330066523765864364137, −14.74267089543101576541555895530, −12.93390188328067123286875432041, −11.79485079053922825350384158510, −10.45800531942356156569227015167, −9.271126911313183201440722262339, −8.436677701926446013822755554984, −7.31463830315092065478924853063, −5.19172451579117052389275741572, −3.76791033714092301790056983077, 0.48630789610858371369796030377, 2.77574176632216723569016484599, 5.43311354085342690422821730573, 7.40775714257113906055881209963, 8.003409386617372521079538156617, 9.496682051306506609732415165860, 11.04783551785735903959636708851, 11.43171758595869132349972551512, 12.93847674817465002511588189938, 14.07437543587214258636299084482

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