↑

Properties

Label 2-63-63.25-c1-0-3
Degree $2$
Conductor $63$
Sign $0.995 + 0.0984i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.670·2-s + (1.65 − 0.518i)3-s − 1.55·4-s + (−0.712 + 1.23i)5-s + (1.10 − 0.347i)6-s + (−2.36 − 1.19i)7-s − 2.38·8-s + (2.46 − 1.71i)9-s + (−0.477 + 0.827i)10-s + (2.46 + 4.27i)11-s + (−2.56 + 0.803i)12-s + (−1.37 − 2.38i)13-s + (−1.58 − 0.801i)14-s + (−0.537 + 2.40i)15-s + 1.50·16-s + (0.559 − 0.969i)17-s + ⋯
L(s)  = 1  + 0.473·2-s + (0.954 − 0.299i)3-s − 0.775·4-s + (−0.318 + 0.551i)5-s + (0.452 − 0.141i)6-s + (−0.892 − 0.451i)7-s − 0.841·8-s + (0.820 − 0.571i)9-s + (−0.151 + 0.261i)10-s + (0.743 + 1.28i)11-s + (−0.739 + 0.232i)12-s + (−0.381 − 0.661i)13-s + (−0.422 − 0.214i)14-s + (−0.138 + 0.621i)15-s + 0.376·16-s + (0.135 − 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.995 + 0.0984i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.995 + 0.0984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10671 - 0.0546212i\)
\(L(\frac12)\) \(\approx\) \(1.10671 - 0.0546212i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 + 0.518i)T \)
7 \( 1 + (2.36 + 1.19i)T \)
good2 \( 1 - 0.670T + 2T^{2} \)
5 \( 1 + (0.712 - 1.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.46 - 4.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.37 + 2.38i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.559 + 0.969i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.00 + 3.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.71 - 4.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.40 + 5.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 + (-0.709 - 1.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.124 - 0.215i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.498 - 0.863i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.47T + 47T^{2} \)
53 \( 1 + (0.410 - 0.710i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.58T + 59T^{2} \)
61 \( 1 - 0.0752T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 - 0.0804T + 71T^{2} \)
73 \( 1 + (-5.34 + 9.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.84T + 79T^{2} \)
83 \( 1 + (7.23 - 12.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.76 - 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.70 + 4.67i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.88579148856781839038277687592, −13.80468048652045218888371864416, −13.00880912820948527608084956559, −12.06368562985963985920813198377, −10.00794681963744098352112849156, −9.316390303837076915638453190609, −7.74289262275230473894488005739, −6.60417417924657211050633381433, −4.38322358029979438503815601339, −3.13888849535205387440684217146, 3.31792143020839315790164686836, 4.49149929585291352046660725836, 6.25303642141925168021465339098, 8.427270012577269147634052007347, 8.937312052733327295470126189990, 10.12057703194864936111701264121, 12.10734435476300503387660587489, 12.90914839223722839686657892911, 14.04704188495754188415888587754, 14.63718811723843953322829691458

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