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Properties

Label 2-63-7.4-c9-0-11
Degree $2$
Conductor $63$
Sign $-0.839 - 0.542i$
Analytic cond. $32.4472$
Root an. cond. $5.69624$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (15.9 + 27.5i)2-s + (−250. + 434. i)4-s + (502. + 870. i)5-s + (3.10e3 − 5.54e3i)7-s + 332.·8-s + (−1.60e4 + 2.77e4i)10-s + (−3.79e4 + 6.57e4i)11-s + 1.66e5·13-s + (2.02e5 − 2.70e3i)14-s + (1.33e5 + 2.31e5i)16-s + (−2.17e5 + 3.76e5i)17-s + (−2.24e5 − 3.88e5i)19-s − 5.04e5·20-s − 2.41e6·22-s + (8.21e5 + 1.42e6i)23-s + ⋯
L(s)  = 1  + (0.703 + 1.21i)2-s + (−0.489 + 0.848i)4-s + (0.359 + 0.622i)5-s + (0.488 − 0.872i)7-s + 0.0286·8-s + (−0.505 + 0.876i)10-s + (−0.781 + 1.35i)11-s + 1.61·13-s + (1.40 − 0.0188i)14-s + (0.509 + 0.883i)16-s + (−0.630 + 1.09i)17-s + (−0.395 − 0.684i)19-s − 0.704·20-s − 2.19·22-s + (0.611 + 1.05i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.839 - 0.542i$
Analytic conductor: \(32.4472\)
Root analytic conductor: \(5.69624\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :9/2),\ -0.839 - 0.542i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.933940 + 3.16666i\)
\(L(\frac12)\) \(\approx\) \(0.933940 + 3.16666i\)
\(L(\frac{11}{2})\) 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 \( 1 + (-3.10e3 + 5.54e3i)T \)
good2 \( 1 + (-15.9 - 27.5i)T + (-256 + 443. i)T^{2} \)
5 \( 1 + (-502. - 870. i)T + (-9.76e5 + 1.69e6i)T^{2} \)
11 \( 1 + (3.79e4 - 6.57e4i)T + (-1.17e9 - 2.04e9i)T^{2} \)
13 \( 1 - 1.66e5T + 1.06e10T^{2} \)
17 \( 1 + (2.17e5 - 3.76e5i)T + (-5.92e10 - 1.02e11i)T^{2} \)
19 \( 1 + (2.24e5 + 3.88e5i)T + (-1.61e11 + 2.79e11i)T^{2} \)
23 \( 1 + (-8.21e5 - 1.42e6i)T + (-9.00e11 + 1.55e12i)T^{2} \)
29 \( 1 + 3.82e6T + 1.45e13T^{2} \)
31 \( 1 + (8.39e5 - 1.45e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (-6.88e6 - 1.19e7i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 - 6.02e5T + 3.27e14T^{2} \)
43 \( 1 + 2.34e7T + 5.02e14T^{2} \)
47 \( 1 + (2.08e7 + 3.60e7i)T + (-5.59e14 + 9.69e14i)T^{2} \)
53 \( 1 + (-2.05e7 + 3.56e7i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 + (2.32e7 - 4.02e7i)T + (-4.33e15 - 7.50e15i)T^{2} \)
61 \( 1 + (-8.99e7 - 1.55e8i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (5.56e7 - 9.64e7i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 - 1.92e8T + 4.58e16T^{2} \)
73 \( 1 + (-1.82e8 + 3.15e8i)T + (-2.94e16 - 5.09e16i)T^{2} \)
79 \( 1 + (1.68e7 + 2.91e7i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 - 2.52e8T + 1.86e17T^{2} \)
89 \( 1 + (-3.67e8 - 6.36e8i)T + (-1.75e17 + 3.03e17i)T^{2} \)
97 \( 1 - 4.21e7T + 7.60e17T^{2} \)
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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

−13.44656248308386126169982232715, −13.20039002419987091871272267701, −11.08091130188965020733351616324, −10.26395274386050082295087375991, −8.362329008578001792239879050499, −7.21311771133954201314193209687, −6.39525127900641999893699542652, −5.01107509621570934476162553667, −3.84256189941213004713636059086, −1.69671932003328070679797297740, 0.827268061713329168778232990168, 2.13742216108495019076655249125, 3.38913231559792340693660649800, 4.91714410669609246312389915181, 5.90575903508278056654561598570, 8.223143016077723730318800795078, 9.196420082102448773836500861854, 10.93573672708974484779153768621, 11.30628062566132135547380333762, 12.74181451523957222780400473078

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