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Properties

Label 2-63-63.47-c3-0-17
Degree $2$
Conductor $63$
Sign $0.768 + 0.639i$
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  + (3.55 − 2.05i)2-s + (4.91 + 1.67i)3-s + (4.43 − 7.68i)4-s − 7.61·5-s + (20.9 − 4.14i)6-s + (4.76 − 17.8i)7-s − 3.56i·8-s + (21.3 + 16.4i)9-s + (−27.0 + 15.6i)10-s + 7.17i·11-s + (34.6 − 30.3i)12-s + (−59.6 + 34.4i)13-s + (−19.7 − 73.4i)14-s + (−37.4 − 12.7i)15-s + (28.1 + 48.7i)16-s + (11.4 + 19.8i)17-s + ⋯
L(s)  = 1  + (1.25 − 0.726i)2-s + (0.946 + 0.322i)3-s + (0.554 − 0.960i)4-s − 0.680·5-s + (1.42 − 0.281i)6-s + (0.257 − 0.966i)7-s − 0.157i·8-s + (0.792 + 0.610i)9-s + (−0.856 + 0.494i)10-s + 0.196i·11-s + (0.834 − 0.730i)12-s + (−1.27 + 0.734i)13-s + (−0.377 − 1.40i)14-s + (−0.644 − 0.219i)15-s + (0.439 + 0.761i)16-s + (0.163 + 0.282i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.768 + 0.639i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.768 + 0.639i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.86508 - 1.03682i\)
\(L(\frac12)\) \(\approx\) \(2.86508 - 1.03682i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.91 - 1.67i)T \)
7 \( 1 + (-4.76 + 17.8i)T \)
good2 \( 1 + (-3.55 + 2.05i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + 7.61T + 125T^{2} \)
11 \( 1 - 7.17iT - 1.33e3T^{2} \)
13 \( 1 + (59.6 - 34.4i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-11.4 - 19.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (86.1 + 49.7i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 122. iT - 1.21e4T^{2} \)
29 \( 1 + (-143. - 82.8i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (11.9 + 6.89i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-126. + 219. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-159. - 276. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-156. + 271. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (35.3 + 61.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-254. + 146. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-230. + 399. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-573. + 331. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (359. - 622. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 92.9iT - 3.57e5T^{2} \)
73 \( 1 + (882. - 509. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (398. + 689. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-10.7 + 18.5i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (339. - 588. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (984. + 568. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.47571406646676386546501151646, −13.25463512122954339993358137786, −12.35658781429544625258676094182, −11.10939051700528639428489775986, −10.07404981314891087881309579508, −8.396890016027267866520966648741, −7.09419694039915948806219417551, −4.63528650021564792379544905247, −4.01492087075896617182499496490, −2.38946354187092569718126446827, 2.82953816356643285908239242932, 4.32698910553452494596992697492, 5.81218976215689620022564420357, 7.34722652795042111913897060035, 8.228382067797471330322857687108, 9.772339734346017996822651128901, 11.92659253104892014483833428257, 12.57552197472699564277154158974, 13.67490137920223993073215168574, 14.80660323029227869577171628329

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