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Properties

Label 2-63-63.59-c3-0-1
Degree $2$
Conductor $63$
Sign $0.425 - 0.904i$
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  + (−4.53 − 2.61i)2-s + (2.85 − 4.34i)3-s + (9.72 + 16.8i)4-s − 12.2·5-s + (−24.3 + 12.2i)6-s + (−10.8 + 15.0i)7-s − 59.9i·8-s + (−10.7 − 24.7i)9-s + (55.3 + 31.9i)10-s + 63.6i·11-s + (100. + 5.81i)12-s + (13.7 + 7.96i)13-s + (88.5 − 39.6i)14-s + (−34.8 + 53.0i)15-s + (−79.2 + 137. i)16-s + (17.9 − 31.0i)17-s + ⋯
L(s)  = 1  + (−1.60 − 0.926i)2-s + (0.549 − 0.835i)3-s + (1.21 + 2.10i)4-s − 1.09·5-s + (−1.65 + 0.832i)6-s + (−0.585 + 0.810i)7-s − 2.64i·8-s + (−0.397 − 0.917i)9-s + (1.75 + 1.01i)10-s + 1.74i·11-s + (2.42 + 0.139i)12-s + (0.294 + 0.169i)13-s + (1.69 − 0.757i)14-s + (−0.599 + 0.912i)15-s + (−1.23 + 2.14i)16-s + (0.255 − 0.443i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.425 - 0.904i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.425 - 0.904i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.201258 + 0.127746i\)
\(L(\frac12)\) \(\approx\) \(0.201258 + 0.127746i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.85 + 4.34i)T \)
7 \( 1 + (10.8 - 15.0i)T \)
good2 \( 1 + (4.53 + 2.61i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 12.2T + 125T^{2} \)
11 \( 1 - 63.6iT - 1.33e3T^{2} \)
13 \( 1 + (-13.7 - 7.96i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-17.9 + 31.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (13.8 - 7.97i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 45.6iT - 1.21e4T^{2} \)
29 \( 1 + (206. - 119. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (117. - 67.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-18.0 - 31.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (84.5 - 146. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (37.8 + 65.6i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-3.46 + 6.00i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (33.2 + 19.2i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (62.2 + 107. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (445. + 257. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (167. + 290. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 716. iT - 3.57e5T^{2} \)
73 \( 1 + (-725. - 418. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-204. + 354. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-135. - 234. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (717. + 1.24e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-433. + 250. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.96630839550113575977471968648, −12.81130355352729553085034653011, −12.22357372316542387031878403417, −11.39358593838241161651087557076, −9.747613415977726819233180254098, −8.932192766298827822537047205610, −7.77040781089237772028931550559, −7.00996125422415941046549272498, −3.41829308870730459343269955215, −1.90726326877764071347297839444, 0.24848209145774819330775981601, 3.66637179837708560903464552573, 5.96192106331141259769580243786, 7.57530313436412866774280878319, 8.326191079712548188593230324202, 9.319477567568412453055575986667, 10.58318206891240229906732863971, 11.19078848175606166620935710076, 13.62970348546430865761904275854, 14.86935953625159720240845733346

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