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Properties

Label 2-63-7.2-c5-0-6
Degree $2$
Conductor $63$
Sign $-0.266 - 0.963i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.59 + 7.95i)2-s + (−26.1 − 45.2i)4-s + (11.0 − 19.1i)5-s + (126. − 26.6i)7-s + 186.·8-s + (101. + 175. i)10-s + (208. + 360. i)11-s + 797.·13-s + (−370. + 1.13e3i)14-s + (−18.6 + 32.3i)16-s + (−687. − 1.19e3i)17-s + (−1.15e3 + 2.00e3i)19-s − 1.15e3·20-s − 3.82e3·22-s + (−477. + 827. i)23-s + ⋯
L(s)  = 1  + (−0.811 + 1.40i)2-s + (−0.817 − 1.41i)4-s + (0.197 − 0.341i)5-s + (0.978 − 0.205i)7-s + 1.02·8-s + (0.320 + 0.554i)10-s + (0.519 + 0.899i)11-s + 1.30·13-s + (−0.505 + 1.54i)14-s + (−0.0182 + 0.0315i)16-s + (−0.577 − 0.999i)17-s + (−0.734 + 1.27i)19-s − 0.645·20-s − 1.68·22-s + (−0.188 + 0.326i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.759666 + 0.998539i\)
\(L(\frac12)\) \(\approx\) \(0.759666 + 0.998539i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-126. + 26.6i)T \)
good2 \( 1 + (4.59 - 7.95i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (-11.0 + 19.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-208. - 360. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 797.T + 3.71e5T^{2} \)
17 \( 1 + (687. + 1.19e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.15e3 - 2.00e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (477. - 827. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 7.03e3T + 2.05e7T^{2} \)
31 \( 1 + (630. + 1.09e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (4.88e3 - 8.46e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 5.40e3T + 1.15e8T^{2} \)
43 \( 1 - 1.96e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.02e3 + 1.78e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-9.01e3 - 1.56e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-3.71e3 - 6.43e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.74e3 - 3.02e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (7.92e3 + 1.37e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 5.81e4T + 1.80e9T^{2} \)
73 \( 1 + (1.95e4 + 3.38e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (4.88e3 - 8.45e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 7.03e4T + 3.93e9T^{2} \)
89 \( 1 + (-7.21e4 + 1.24e5i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 7.93e4T + 8.58e9T^{2} \)
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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

−14.57689462076721489227273039849, −13.67429754556134512360356435192, −12.01063737435377176942895124473, −10.53021928361379567888608450535, −9.185959132174689573027062286562, −8.332815347032031813548793087470, −7.20490819435866488590318732896, −5.96997733860352838886881321955, −4.55324412657194406911919212732, −1.28391182798183293440568151850, 0.972229520143510957034566323779, 2.43726799957064450052214165115, 4.07392881747717945016032973105, 6.27644831174362340722738122564, 8.434783798853305001853379696071, 8.877450364585415585536648421823, 10.75829887240171477842864704976, 10.90331163504154518327586012205, 12.13396741736781814044365324209, 13.37680530599984048403124260829

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