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Properties

Label 2-63-63.38-c3-0-19
Degree $2$
Conductor $63$
Sign $-0.963 - 0.269i$
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  − 4.92i·2-s + (3.60 − 3.74i)3-s − 16.2·4-s + (−4.62 + 8.00i)5-s + (−18.4 − 17.7i)6-s + (−13.0 − 13.1i)7-s + 40.8i·8-s + (−1.04 − 26.9i)9-s + (39.4 + 22.7i)10-s + (13.2 − 7.62i)11-s + (−58.6 + 61.0i)12-s + (61.0 − 35.2i)13-s + (−64.6 + 64.4i)14-s + (13.3 + 46.1i)15-s + 71.0·16-s + (−17.8 + 30.9i)17-s + ⋯
L(s)  = 1  − 1.74i·2-s + (0.693 − 0.720i)3-s − 2.03·4-s + (−0.413 + 0.716i)5-s + (−1.25 − 1.20i)6-s + (−0.705 − 0.708i)7-s + 1.80i·8-s + (−0.0388 − 0.999i)9-s + (1.24 + 0.720i)10-s + (0.362 − 0.209i)11-s + (−1.41 + 1.46i)12-s + (1.30 − 0.752i)13-s + (−1.23 + 1.22i)14-s + (0.229 + 0.794i)15-s + 1.11·16-s + (−0.255 + 0.441i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.963 - 0.269i$
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: \((2,\ 63,\ (\ :3/2),\ -0.963 - 0.269i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.183317 + 1.33625i\)
\(L(\frac12)\) \(\approx\) \(0.183317 + 1.33625i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.60 + 3.74i)T \)
7 \( 1 + (13.0 + 13.1i)T \)
good2 \( 1 + 4.92iT - 8T^{2} \)
5 \( 1 + (4.62 - 8.00i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-13.2 + 7.62i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-61.0 + 35.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (17.8 - 30.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-91.6 + 52.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (80.0 + 46.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-222. - 128. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 142. iT - 2.97e4T^{2} \)
37 \( 1 + (100. + 173. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-162. - 280. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (92.5 - 160. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 471.T + 1.03e5T^{2} \)
53 \( 1 + (-247. - 142. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 170.T + 2.05e5T^{2} \)
61 \( 1 + 257. iT - 2.26e5T^{2} \)
67 \( 1 - 1.00e3T + 3.00e5T^{2} \)
71 \( 1 - 445. iT - 3.57e5T^{2} \)
73 \( 1 + (81.3 + 46.9i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 463.T + 4.93e5T^{2} \)
83 \( 1 + (-348. + 603. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-338. - 586. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (402. + 232. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.46604917321703539458675201963, −12.78356405990745669232894838420, −11.54385147940049778209312620479, −10.63707719205482679509464709540, −9.484933696180464461111915275620, −8.200482442004087583407371018944, −6.62250865333260432977702555837, −3.76620412730336584937729743124, −2.98564752038031549584017405310, −0.961531433778671484488781829017, 3.87889297453154010059920558523, 5.20534466885961021332858297450, 6.58188694613211172157941476548, 8.174327385768024242435480162931, 8.837599399033942712169725985745, 9.770215273832517454038322349583, 11.95158457174129282827444098282, 13.51541743889093187074878193213, 14.17494981569733620772415237621, 15.48096783642901426272708258050

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