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Properties

Label 2-63-63.20-c3-0-8
Degree $2$
Conductor $63$
Sign $-0.331 - 0.943i$
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  + (3.92 + 2.26i)2-s + (−3.93 + 3.39i)3-s + (6.27 + 10.8i)4-s + (0.0687 + 0.119i)5-s + (−23.1 + 4.38i)6-s + (2.53 + 18.3i)7-s + 20.6i·8-s + (4.00 − 26.7i)9-s + 0.623i·10-s + (8.78 + 5.06i)11-s + (−61.5 − 21.5i)12-s + (13.1 − 7.57i)13-s + (−31.6 + 77.7i)14-s + (−0.674 − 0.235i)15-s + (3.45 − 5.98i)16-s + 88.2·17-s + ⋯
L(s)  = 1  + (1.38 + 0.801i)2-s + (−0.757 + 0.652i)3-s + (0.784 + 1.35i)4-s + (0.00614 + 0.0106i)5-s + (−1.57 + 0.298i)6-s + (0.137 + 0.990i)7-s + 0.911i·8-s + (0.148 − 0.988i)9-s + 0.0197i·10-s + (0.240 + 0.138i)11-s + (−1.48 − 0.517i)12-s + (0.279 − 0.161i)13-s + (−0.603 + 1.48i)14-s + (−0.0116 − 0.00405i)15-s + (0.0539 − 0.0934i)16-s + 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.331 - 0.943i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.331 - 0.943i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.34978 + 1.90452i\)
\(L(\frac12)\) \(\approx\) \(1.34978 + 1.90452i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.93 - 3.39i)T \)
7 \( 1 + (-2.53 - 18.3i)T \)
good2 \( 1 + (-3.92 - 2.26i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-0.0687 - 0.119i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-8.78 - 5.06i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-13.1 + 7.57i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 88.2T + 4.91e3T^{2} \)
19 \( 1 + 66.1iT - 6.85e3T^{2} \)
23 \( 1 + (49.7 - 28.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-102. - 59.3i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (266. - 153. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + (220. + 382. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-46.7 + 80.9i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-155. + 270. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 321. iT - 1.48e5T^{2} \)
59 \( 1 + (-287. - 497. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (634. + 366. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-31.6 - 54.7i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 621. iT - 3.57e5T^{2} \)
73 \( 1 + 95.2iT - 3.89e5T^{2} \)
79 \( 1 + (-425. + 736. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (70.1 - 121. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.22e3T + 7.04e5T^{2} \)
97 \( 1 + (-671. - 387. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.90320977427550817006754514286, −13.97812723260934119696251176456, −12.40002172939431087399724366298, −12.01900789334782008023962538280, −10.46062049564969588162261916995, −8.875728836656458253517072471903, −7.00563429929800435622825012150, −5.76613789081171206444716927855, −5.00920626692116031972735680000, −3.47834195135017849576314158458, 1.46164934280198041526705875009, 3.65559967463990772983126112288, 5.10265663523549491540250095692, 6.31180228137229184489947329980, 7.77645590572974413610168320142, 10.21205457239376603319668648760, 11.12943076718874724785582742287, 12.04527173474040973548842150487, 12.94479243665617179840324945862, 13.83957365095452554470989608964

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