Properties

Label 2-63-63.32-c2-0-11
Degree $2$
Conductor $63$
Sign $-0.545 + 0.837i$
Analytic cond. $1.71662$
Root an. cond. $1.31020$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0664 + 0.0383i)2-s + (−2.92 + 0.660i)3-s + (−1.99 − 3.45i)4-s − 4.07i·5-s + (−0.219 − 0.0684i)6-s + (−6.97 − 0.595i)7-s − 0.613i·8-s + (8.12 − 3.86i)9-s + (0.156 − 0.270i)10-s − 12.4i·11-s + (8.12 + 8.80i)12-s + (−5.97 + 10.3i)13-s + (−0.440 − 0.307i)14-s + (2.69 + 11.9i)15-s + (−7.96 + 13.7i)16-s + (14.2 + 8.24i)17-s + ⋯
L(s)  = 1  + (0.0332 + 0.0191i)2-s + (−0.975 + 0.220i)3-s + (−0.499 − 0.864i)4-s − 0.815i·5-s + (−0.0366 − 0.0114i)6-s + (−0.996 − 0.0850i)7-s − 0.0766i·8-s + (0.903 − 0.429i)9-s + (0.0156 − 0.0270i)10-s − 1.13i·11-s + (0.677 + 0.733i)12-s + (−0.459 + 0.796i)13-s + (−0.0314 − 0.0219i)14-s + (0.179 + 0.795i)15-s + (−0.497 + 0.862i)16-s + (0.840 + 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.837i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.545 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.545 + 0.837i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ -0.545 + 0.837i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.269493 - 0.497273i\)
\(L(\frac12)\) \(\approx\) \(0.269493 - 0.497273i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.92 - 0.660i)T \)
7 \( 1 + (6.97 + 0.595i)T \)
good2 \( 1 + (-0.0664 - 0.0383i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + 4.07iT - 25T^{2} \)
11 \( 1 + 12.4iT - 121T^{2} \)
13 \( 1 + (5.97 - 10.3i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-14.2 - 8.24i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (3.69 + 6.39i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + 27.9iT - 529T^{2} \)
29 \( 1 + (-32.2 + 18.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (16.2 + 28.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (10.3 + 17.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (26.1 + 15.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (12.7 + 22.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-68.4 - 39.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (61.5 + 35.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (37.3 - 21.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-11.2 + 19.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-43.0 - 74.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 102. iT - 5.04e3T^{2} \)
73 \( 1 + (-0.403 + 0.699i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-13.5 + 23.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-36.4 + 21.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-36.1 + 20.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-6.66 - 11.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−14.28702425871908376255493182539, −13.11303988243103836299450020733, −12.19832787321939633214325362049, −10.79179444103019010285664581986, −9.815678255879855409240738339495, −8.790895504557156187105724241231, −6.55040198108333176437374397279, −5.57391262667737390778022705655, −4.26692191168672387312777512371, −0.59413969119455105786545332309, 3.22792206174740307699393402859, 5.07689049621570319676630843405, 6.75830482095282819616459736302, 7.61759677068337380191035326903, 9.599847199042161627083508872572, 10.51578956029609467015897140816, 12.13997062480767320358331881146, 12.53875543258654317230730651818, 13.75845001542822686094492674134, 15.24393841678684512856036430714

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