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Properties

Label 2-63-63.20-c3-0-20
Degree $2$
Conductor $63$
Sign $-0.999 + 0.0112i$
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  + (−1.10 − 0.638i)2-s + (0.213 − 5.19i)3-s + (−3.18 − 5.51i)4-s + (−1.59 − 2.76i)5-s + (−3.55 + 5.60i)6-s + (−4.68 + 17.9i)7-s + 18.3i·8-s + (−26.9 − 2.21i)9-s + 4.07i·10-s + (−38.0 − 21.9i)11-s + (−29.3 + 15.3i)12-s + (65.6 − 37.8i)13-s + (16.6 − 16.8i)14-s + (−14.6 + 7.69i)15-s + (−13.7 + 23.8i)16-s − 104.·17-s + ⋯
L(s)  = 1  + (−0.390 − 0.225i)2-s + (0.0411 − 0.999i)3-s + (−0.398 − 0.689i)4-s + (−0.142 − 0.247i)5-s + (−0.241 + 0.381i)6-s + (−0.252 + 0.967i)7-s + 0.810i·8-s + (−0.996 − 0.0821i)9-s + 0.128i·10-s + (−1.04 − 0.602i)11-s + (−0.705 + 0.369i)12-s + (1.39 − 0.808i)13-s + (0.317 − 0.321i)14-s + (−0.252 + 0.132i)15-s + (−0.215 + 0.372i)16-s − 1.49·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0112i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.999 + 0.0112i$
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: \((2,\ 63,\ (\ :3/2),\ -0.999 + 0.0112i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00360356 - 0.639502i\)
\(L(\frac12)\) \(\approx\) \(0.00360356 - 0.639502i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.213 + 5.19i)T \)
7 \( 1 + (4.68 - 17.9i)T \)
good2 \( 1 + (1.10 + 0.638i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (1.59 + 2.76i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (38.0 + 21.9i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-65.6 + 37.8i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 104.T + 4.91e3T^{2} \)
19 \( 1 + 74.7iT - 6.85e3T^{2} \)
23 \( 1 + (-46.6 + 26.9i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (26.3 + 15.2i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-111. + 64.2i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 46.1T + 5.06e4T^{2} \)
41 \( 1 + (213. + 369. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (166. - 287. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-170. + 294. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 235. iT - 1.48e5T^{2} \)
59 \( 1 + (-272. - 471. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-321. - 185. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (53.3 + 92.4i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 974. iT - 3.57e5T^{2} \)
73 \( 1 + 576. iT - 3.89e5T^{2} \)
79 \( 1 + (10.6 - 18.4i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-168. + 292. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + (63.5 + 36.6i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.47330684129726735711965935230, −13.11122465730207243089215239092, −11.53956173540769639600909287288, −10.62297014890498164903448620681, −8.792753639942182877041697493836, −8.418946226075881925383957024845, −6.38715811262463399351900205941, −5.31346056261729857504202182518, −2.51117974182639297785131074557, −0.49345209124960186739936645466, 3.48327868966045589279621340833, 4.57407002390119738146104874945, 6.72929128689675794064780090437, 8.115841948236206769383827039974, 9.201567766728909369733549691094, 10.35660030962384432491939076703, 11.30951409012076434224372006059, 13.04353795586894199362960068135, 13.83913858968480891686332949483, 15.35926196088010865604396238132

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