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Properties

Label 2-63-63.38-c1-0-4
Degree $2$
Conductor $63$
Sign $-0.122 + 0.992i$
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  − 2.59i·2-s + (1.34 + 1.09i)3-s − 4.72·4-s + (0.626 − 1.08i)5-s + (2.84 − 3.47i)6-s + (−1.89 + 1.84i)7-s + 7.07i·8-s + (0.599 + 2.93i)9-s + (−2.81 − 1.62i)10-s + (−0.534 + 0.308i)11-s + (−6.34 − 5.17i)12-s + (1.06 − 0.613i)13-s + (4.78 + 4.92i)14-s + (2.02 − 0.769i)15-s + 8.88·16-s + (2.21 − 3.83i)17-s + ⋯
L(s)  = 1  − 1.83i·2-s + (0.774 + 0.632i)3-s − 2.36·4-s + (0.280 − 0.485i)5-s + (1.16 − 1.42i)6-s + (−0.717 + 0.696i)7-s + 2.50i·8-s + (0.199 + 0.979i)9-s + (−0.889 − 0.513i)10-s + (−0.161 + 0.0929i)11-s + (−1.83 − 1.49i)12-s + (0.294 − 0.170i)13-s + (1.27 + 1.31i)14-s + (0.523 − 0.198i)15-s + 2.22·16-s + (0.537 − 0.930i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.122 + 0.992i$
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: \((2,\ 63,\ (\ :1/2),\ -0.122 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.631977 - 0.714734i\)
\(L(\frac12)\) \(\approx\) \(0.631977 - 0.714734i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.34 - 1.09i)T \)
7 \( 1 + (1.89 - 1.84i)T \)
good2 \( 1 + 2.59iT - 2T^{2} \)
5 \( 1 + (-0.626 + 1.08i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.534 - 0.308i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.06 + 0.613i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.21 + 3.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.64 - 0.950i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.11 + 2.37i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.07 + 2.93i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.48iT - 31T^{2} \)
37 \( 1 + (-1.33 - 2.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.09 + 3.63i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.24 - 3.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.61T + 47T^{2} \)
53 \( 1 + (2.67 + 1.54i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 3.56T + 59T^{2} \)
61 \( 1 + 14.4iT - 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (-9.95 - 5.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.03T + 79T^{2} \)
83 \( 1 + (4.36 - 7.56i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.811 - 1.40i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.76 - 5.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.26118856975441911087095115445, −13.24825193753315340477253426123, −12.49784800175515714895548963143, −11.23722918398700089411391213770, −9.938354692208885847072767682546, −9.406325629227750639468486772485, −8.339534536834143685160529088486, −5.24996934162755556475000929905, −3.72548420776915063705268294546, −2.38689141958865116276646550313, 3.80392450691408923662945055485, 6.00749317218271082949108148903, 6.88988462070366318125480203112, 7.85983545816181433438674536387, 8.993459269673165552108802506432, 10.20610073482877297815834451708, 12.68613160375460222643435963050, 13.58366561354356241786845569736, 14.29231308923270525622676630516, 15.16588689974219946434001755978

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