Properties

Label 2-63-7.4-c5-0-9
Degree $2$
Conductor $63$
Sign $0.962 - 0.273i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.44 + 4.23i)2-s + (4.04 − 7.00i)4-s + (−21.3 − 37.0i)5-s + (43.2 + 122. i)7-s + 196.·8-s + (104. − 181. i)10-s + (355. − 615. i)11-s + 885.·13-s + (−411. + 481. i)14-s + (349. + 605. i)16-s + (−350. + 607. i)17-s + (627. + 1.08e3i)19-s − 345.·20-s + 3.47e3·22-s + (−523. − 906. i)23-s + ⋯
L(s)  = 1  + (0.432 + 0.748i)2-s + (0.126 − 0.219i)4-s + (−0.382 − 0.662i)5-s + (0.333 + 0.942i)7-s + 1.08·8-s + (0.330 − 0.572i)10-s + (0.885 − 1.53i)11-s + 1.45·13-s + (−0.561 + 0.657i)14-s + (0.341 + 0.591i)16-s + (−0.294 + 0.509i)17-s + (0.399 + 0.691i)19-s − 0.193·20-s + 1.53·22-s + (−0.206 − 0.357i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.273i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.962 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.962 - 0.273i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ 0.962 - 0.273i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.46518 + 0.343071i\)
\(L(\frac12)\) \(\approx\) \(2.46518 + 0.343071i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-43.2 - 122. i)T \)
good2 \( 1 + (-2.44 - 4.23i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (21.3 + 37.0i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-355. + 615. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 885.T + 3.71e5T^{2} \)
17 \( 1 + (350. - 607. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-627. - 1.08e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (523. + 906. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 6.15e3T + 2.05e7T^{2} \)
31 \( 1 + (1.14e3 - 1.98e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (202. + 349. i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.78e4T + 1.15e8T^{2} \)
43 \( 1 + 1.46e4T + 1.47e8T^{2} \)
47 \( 1 + (1.05e4 + 1.83e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-53.6 + 92.8i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (2.22e4 - 3.85e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.16e4 - 2.01e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-3.33e3 + 5.77e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 2.51e4T + 1.80e9T^{2} \)
73 \( 1 + (4.73e3 - 8.20e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.32e4 - 2.29e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 7.49e3T + 3.93e9T^{2} \)
89 \( 1 + (-1.60e4 - 2.78e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.55e5T + 8.58e9T^{2} \)
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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

−14.18703418341281853157257772954, −13.16321406173146882177409846879, −11.75364389669332166445266803199, −10.85255191912472684043985234358, −8.918889748065942793149460955614, −8.159527127957678237781930904481, −6.29516450969756291892880494783, −5.55621266696825383838971253202, −3.90166697080233916253372226402, −1.31221771038398314558836425364, 1.59873024521014859638442577510, 3.46680357121096204315094465715, 4.45451588099477670492902268806, 6.85728818047799667854822832658, 7.66259098694114704382780806407, 9.564863210929920070368711498560, 11.03933382684486204304738083334, 11.38222932113276359705223394291, 12.78574369894699390843414107736, 13.72296086313202294928086606342

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