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Properties

Label 2-63-63.38-c1-0-1
Degree $2$
Conductor $63$
Sign $0.995 - 0.0917i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.718i·2-s + (−0.271 + 1.71i)3-s + 1.48·4-s + (−0.723 + 1.25i)5-s + (1.22 + 0.194i)6-s + (0.182 − 2.63i)7-s − 2.50i·8-s + (−2.85 − 0.928i)9-s + (0.900 + 0.519i)10-s + (−1.55 + 0.900i)11-s + (−0.402 + 2.53i)12-s + (−1.88 + 1.09i)13-s + (−1.89 − 0.131i)14-s + (−1.94 − 1.57i)15-s + 1.17·16-s + (1.95 − 3.38i)17-s + ⋯
L(s)  = 1  − 0.507i·2-s + (−0.156 + 0.987i)3-s + 0.742·4-s + (−0.323 + 0.560i)5-s + (0.501 + 0.0795i)6-s + (0.0690 − 0.997i)7-s − 0.884i·8-s + (−0.950 − 0.309i)9-s + (0.284 + 0.164i)10-s + (−0.470 + 0.271i)11-s + (−0.116 + 0.732i)12-s + (−0.523 + 0.302i)13-s + (−0.506 − 0.0350i)14-s + (−0.502 − 0.407i)15-s + 0.292·16-s + (0.473 − 0.820i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.995 - 0.0917i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.995 - 0.0917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.914825 + 0.0420690i\)
\(L(\frac12)\) \(\approx\) \(0.914825 + 0.0420690i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.271 - 1.71i)T \)
7 \( 1 + (-0.182 + 2.63i)T \)
good2 \( 1 + 0.718iT - 2T^{2} \)
5 \( 1 + (0.723 - 1.25i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.55 - 0.900i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.88 - 1.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.95 + 3.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.47 - 2.00i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.91 + 2.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.49 - 4.90i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.83iT - 31T^{2} \)
37 \( 1 + (0.411 + 0.713i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.90 - 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.76 - 6.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.33T + 47T^{2} \)
53 \( 1 + (-0.996 - 0.575i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.79T + 59T^{2} \)
61 \( 1 + 2.35iT - 61T^{2} \)
67 \( 1 + 0.312T + 67T^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 + (-2.42 - 1.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + (-3.60 + 6.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.28 + 9.16i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.4 + 7.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−15.02026941983057947543156877926, −14.17068657165005537192028540102, −12.44332527169284560633119138503, −11.33650829746031927175453237562, −10.51957756423327756118641700459, −9.807109603796577538972840290133, −7.79155191601275371019979796625, −6.50215961477640088728777173348, −4.48496838686024492767822769391, −3.02204049518623848062583585208, 2.39052595582068207189811341940, 5.39417103548898093044767210448, 6.40919032910134293614382460462, 7.86554857064290669411125710669, 8.542228722404626247327679687954, 10.65590264917666061779785788281, 12.07486583161329388526711227550, 12.37866947532855836661466599713, 13.92472750419476772380818830785, 15.15134808670102400337966167066

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