↑

Properties

Label 2-504-63.47-c1-0-4
Degree $2$
Conductor $504$
Sign $0.546 - 0.837i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (−1.69 − 0.335i)3-s − 3.64·5-s + (1.05 − 2.42i)7-s + (2.77 + 1.14i)9-s + 1.39i·11-s + (−2.97 + 1.71i)13-s + (6.19 + 1.22i)15-s + (−2.41 − 4.18i)17-s + (7.23 + 4.17i)19-s + (−2.60 + 3.76i)21-s + 8.92i·23-s + 8.29·25-s + (−4.33 − 2.86i)27-s + (5.02 + 2.89i)29-s + (5.29 + 3.05i)31-s + ⋯
L(s)  = 1  + (−0.981 − 0.193i)3-s − 1.63·5-s + (0.398 − 0.917i)7-s + (0.924 + 0.380i)9-s + 0.421i·11-s + (−0.825 + 0.476i)13-s + (1.59 + 0.315i)15-s + (−0.586 − 1.01i)17-s + (1.66 + 0.958i)19-s + (−0.568 + 0.822i)21-s + 1.86i·23-s + 1.65·25-s + (−0.833 − 0.552i)27-s + (0.932 + 0.538i)29-s + (0.951 + 0.549i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.546 - 0.837i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.546 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.520885 + 0.282171i\)
\(L(\frac12)\) \(\approx\) \(0.520885 + 0.282171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.69 + 0.335i)T \)
7 \( 1 + (-1.05 + 2.42i)T \)
good5 \( 1 + 3.64T + 5T^{2} \)
11 \( 1 - 1.39iT - 11T^{2} \)
13 \( 1 + (2.97 - 1.71i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.41 + 4.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.23 - 4.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.92iT - 23T^{2} \)
29 \( 1 + (-5.02 - 2.89i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.29 - 3.05i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.89 - 5.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.802 - 1.38i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.22 + 3.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.51 - 2.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.40 - 2.54i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.21 + 5.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.78 - 4.49i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.53 + 4.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.20iT - 71T^{2} \)
73 \( 1 + (0.745 - 0.430i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.31 + 2.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.82 - 6.62i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.44 - 7.69i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.63 - 5.56i)T + (48.5 + 84.0i)T^{2} \)
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

−11.35019453652944132332052369399, −10.35819920650418017032162215686, −9.478697358189137028999142286861, −7.88608961152710000265050574017, −7.42845101892018253150850542582, −6.81168040447559034503303138578, −5.11543735125966145559650445646, −4.53581926636644348156941735534, −3.42929254983215444815298991371, −1.15040893457989051429532415504, 0.49330012204480889246654299454, 2.84323561560875197325762836395, 4.27913971194090435305058795211, 4.94228100864966871636663609571, 6.08928992310702379049720962147, 7.15384956034626846591648483575, 8.056740320992852354590224380680, 8.853248020415212978853474104098, 10.12390687420909986376134285586, 11.05575505994455652839439902681

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