Properties

Label 2-63-63.25-c3-0-0
Degree $2$
Conductor $63$
Sign $-0.880 - 0.474i$
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  + 1.33·2-s + (−3.51 − 3.82i)3-s − 6.21·4-s + (−4.50 + 7.79i)5-s + (−4.69 − 5.11i)6-s + (−3.16 + 18.2i)7-s − 18.9·8-s + (−2.28 + 26.9i)9-s + (−6.01 + 10.4i)10-s + (−14.6 − 25.4i)11-s + (21.8 + 23.7i)12-s + (−21.1 − 36.6i)13-s + (−4.22 + 24.3i)14-s + (45.6 − 10.1i)15-s + 24.3·16-s + (−2.56 + 4.45i)17-s + ⋯
L(s)  = 1  + 0.472·2-s + (−0.676 − 0.736i)3-s − 0.777·4-s + (−0.402 + 0.697i)5-s + (−0.319 − 0.347i)6-s + (−0.170 + 0.985i)7-s − 0.839·8-s + (−0.0845 + 0.996i)9-s + (−0.190 + 0.329i)10-s + (−0.402 − 0.696i)11-s + (0.525 + 0.572i)12-s + (−0.451 − 0.782i)13-s + (−0.0805 + 0.465i)14-s + (0.786 − 0.175i)15-s + 0.380·16-s + (−0.0366 + 0.0634i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.880 - 0.474i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.880 - 0.474i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0448532 + 0.177856i\)
\(L(\frac12)\) \(\approx\) \(0.0448532 + 0.177856i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.51 + 3.82i)T \)
7 \( 1 + (3.16 - 18.2i)T \)
good2 \( 1 - 1.33T + 8T^{2} \)
5 \( 1 + (4.50 - 7.79i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (14.6 + 25.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (21.1 + 36.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (2.56 - 4.45i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (71.2 + 123. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (89.0 - 154. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (109. - 189. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 147.T + 2.97e4T^{2} \)
37 \( 1 + (21.2 + 36.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-83.7 - 145. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-121. + 210. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 76.5T + 1.03e5T^{2} \)
53 \( 1 + (181. - 314. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 121.T + 2.05e5T^{2} \)
61 \( 1 + 642.T + 2.26e5T^{2} \)
67 \( 1 + 162.T + 3.00e5T^{2} \)
71 \( 1 + 833.T + 3.57e5T^{2} \)
73 \( 1 + (62.4 - 108. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 842.T + 4.93e5T^{2} \)
83 \( 1 + (-566. + 982. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (248. + 429. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (128. - 223. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−14.88320768112430072714414210771, −13.56686916888649680532928287176, −12.79261474334609383614611718044, −11.78024287318539103056919517503, −10.69799409222692369365290519669, −8.996271394843199363757115408367, −7.67052263026035363929714248440, −6.12500130306508122076984194270, −5.10274942083546438571347269870, −2.97374072678103684426180306545, 0.11639259235316931267638510702, 4.14005566453084925193243253357, 4.58397012452068235678712859701, 6.23881276858567029054858067150, 8.140928910832778568954542335994, 9.578414653872218991256725220546, 10.41988335014429130680826567432, 12.05940493262135679924275277381, 12.68908062570878730387813245909, 14.07072421791490849970023145989

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