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Properties

Label 2-63-9.4-c3-0-3
Degree $2$
Conductor $63$
Sign $-0.107 - 0.994i$
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  + (−0.403 − 0.699i)2-s + (−0.172 + 5.19i)3-s + (3.67 − 6.36i)4-s + (−9.11 + 15.7i)5-s + (3.70 − 1.97i)6-s + (3.5 + 6.06i)7-s − 12.3·8-s + (−26.9 − 1.79i)9-s + 14.7·10-s + (24.7 + 42.8i)11-s + (32.4 + 20.1i)12-s + (−22.2 + 38.5i)13-s + (2.82 − 4.89i)14-s + (−80.4 − 50.0i)15-s + (−24.3 − 42.2i)16-s + 47.2·17-s + ⋯
L(s)  = 1  + (−0.142 − 0.247i)2-s + (−0.0331 + 0.999i)3-s + (0.459 − 0.795i)4-s + (−0.815 + 1.41i)5-s + (0.251 − 0.134i)6-s + (0.188 + 0.327i)7-s − 0.547·8-s + (−0.997 − 0.0663i)9-s + 0.465·10-s + (0.677 + 1.17i)11-s + (0.779 + 0.485i)12-s + (−0.474 + 0.821i)13-s + (0.0539 − 0.0934i)14-s + (−1.38 − 0.861i)15-s + (−0.381 − 0.660i)16-s + 0.673·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.107 - 0.994i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.107 - 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.758013 + 0.844788i\)
\(L(\frac12)\) \(\approx\) \(0.758013 + 0.844788i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.172 - 5.19i)T \)
7 \( 1 + (-3.5 - 6.06i)T \)
good2 \( 1 + (0.403 + 0.699i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (9.11 - 15.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-24.7 - 42.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (22.2 - 38.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 47.2T + 4.91e3T^{2} \)
19 \( 1 - 56.7T + 6.85e3T^{2} \)
23 \( 1 + (-27.7 + 47.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (75.1 + 130. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-83.7 + 145. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 331.T + 5.06e4T^{2} \)
41 \( 1 + (147. - 255. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-158. - 275. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-68.9 - 119. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 411.T + 1.48e5T^{2} \)
59 \( 1 + (106. - 183. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-26.1 - 45.3i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (353. - 612. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 78.7T + 3.57e5T^{2} \)
73 \( 1 - 839.T + 3.89e5T^{2} \)
79 \( 1 + (507. + 879. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (543. + 941. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 762.T + 7.04e5T^{2} \)
97 \( 1 + (-671. - 1.16e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.80503386844221414421058216101, −14.40125913487479818721024089085, −11.75592474653098326742214034067, −11.44534241600273436668774603987, −10.16200655633018624962642925968, −9.478512863010013766107025907117, −7.46158962744286486903889973458, −6.18506481759862852768286883151, −4.37819390138831972577296168675, −2.67535155878948797692090943778, 0.855898452752791071067842181422, 3.45090934989475781108865918861, 5.52022059573125348851667993784, 7.23942121421952513681801955072, 8.100907434883404949432634455234, 8.877497746063119316478186582601, 11.28411502360425798655295618644, 12.11766099375838786149493676910, 12.77683075464160987565819345321, 13.94020587693126084577760574882

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