Properties

Label 2-63-63.38-c3-0-13
Degree $2$
Conductor $63$
Sign $0.942 - 0.335i$
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  + 0.747i·2-s + (4.77 + 2.05i)3-s + 7.44·4-s + (4.35 − 7.53i)5-s + (−1.53 + 3.56i)6-s + (−11.6 − 14.4i)7-s + 11.5i·8-s + (18.5 + 19.5i)9-s + (5.63 + 3.25i)10-s + (−35.9 + 20.7i)11-s + (35.5 + 15.2i)12-s + (33.0 − 19.0i)13-s + (10.7 − 8.68i)14-s + (36.2 − 27.0i)15-s + 50.8·16-s + (−39.9 + 69.2i)17-s + ⋯
L(s)  = 1  + 0.264i·2-s + (0.918 + 0.394i)3-s + 0.930·4-s + (0.389 − 0.674i)5-s + (−0.104 + 0.242i)6-s + (−0.627 − 0.778i)7-s + 0.510i·8-s + (0.688 + 0.725i)9-s + (0.178 + 0.102i)10-s + (−0.985 + 0.568i)11-s + (0.854 + 0.367i)12-s + (0.704 − 0.406i)13-s + (0.205 − 0.165i)14-s + (0.623 − 0.465i)15-s + 0.795·16-s + (−0.570 + 0.987i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.942 - 0.335i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.942 - 0.335i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.12588 + 0.367238i\)
\(L(\frac12)\) \(\approx\) \(2.12588 + 0.367238i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.77 - 2.05i)T \)
7 \( 1 + (11.6 + 14.4i)T \)
good2 \( 1 - 0.747iT - 8T^{2} \)
5 \( 1 + (-4.35 + 7.53i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (35.9 - 20.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-33.0 + 19.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (39.9 - 69.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (49.4 - 28.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (153. + 88.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (60.1 + 34.7i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 260. iT - 2.97e4T^{2} \)
37 \( 1 + (74.9 + 129. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-54.7 - 94.7i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-124. + 215. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 295.T + 1.03e5T^{2} \)
53 \( 1 + (263. + 152. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 720.T + 2.05e5T^{2} \)
61 \( 1 - 648. iT - 2.26e5T^{2} \)
67 \( 1 + 194.T + 3.00e5T^{2} \)
71 \( 1 + 98.7iT - 3.57e5T^{2} \)
73 \( 1 + (-226. - 130. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 1.15e3T + 4.93e5T^{2} \)
83 \( 1 + (-285. + 494. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (89.5 + 155. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-956. - 552. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.71755557235785848198812086402, −13.34593049727106486747447408616, −12.70942946625403138177628832012, −10.72743022402654889468903777905, −10.03376846911420127490967607239, −8.497276174989016092249596230283, −7.48485538080292167527194057722, −5.94048042591268153725138479919, −4.02765286990039475001600478699, −2.17403201635985753344483416767, 2.23591213537052486286734000014, 3.20147163017417786354266249979, 6.09031030762297656745562851620, 7.06452279296784414590664419714, 8.519148932329782574188056942370, 9.824714596740955420771800559018, 10.98748040751398559921084011772, 12.24433450836725649509136507180, 13.33837847554443294342554928935, 14.30413831042165842458701708498

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