↑

Properties

Label 2-504-63.5-c1-0-8
Degree $2$
Conductor $504$
Sign $0.771 - 0.635i$
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  + (1.73 − 0.0755i)3-s + (0.684 + 1.18i)5-s + (−1.25 + 2.33i)7-s + (2.98 − 0.261i)9-s + (1.04 + 0.601i)11-s + (−0.639 − 0.369i)13-s + (1.27 + 1.99i)15-s + (0.693 + 1.20i)17-s + (2.81 + 1.62i)19-s + (−1.99 + 4.12i)21-s + (−3.30 + 1.90i)23-s + (1.56 − 2.70i)25-s + (5.15 − 0.677i)27-s + (−3.50 + 2.02i)29-s − 1.18i·31-s + ⋯
L(s)  = 1  + (0.999 − 0.0435i)3-s + (0.305 + 0.529i)5-s + (−0.473 + 0.880i)7-s + (0.996 − 0.0871i)9-s + (0.314 + 0.181i)11-s + (−0.177 − 0.102i)13-s + (0.328 + 0.516i)15-s + (0.168 + 0.291i)17-s + (0.646 + 0.373i)19-s + (−0.434 + 0.900i)21-s + (−0.688 + 0.397i)23-s + (0.312 − 0.541i)25-s + (0.991 − 0.130i)27-s + (−0.649 + 0.375i)29-s − 0.213i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.771 - 0.635i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.771 - 0.635i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91409 + 0.687056i\)
\(L(\frac12)\) \(\approx\) \(1.91409 + 0.687056i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 0.0755i)T \)
7 \( 1 + (1.25 - 2.33i)T \)
good5 \( 1 + (-0.684 - 1.18i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.04 - 0.601i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.639 + 0.369i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.693 - 1.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.81 - 1.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.30 - 1.90i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.50 - 2.02i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.18iT - 31T^{2} \)
37 \( 1 + (-5.10 + 8.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.670 + 1.16i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.490 + 0.848i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.27T + 47T^{2} \)
53 \( 1 + (-5.77 + 3.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 + 5.03iT - 61T^{2} \)
67 \( 1 - 4.39T + 67T^{2} \)
71 \( 1 - 8.84iT - 71T^{2} \)
73 \( 1 + (-10.2 + 5.94i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + (-3.14 - 5.44i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.05 + 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.9 - 6.33i)T + (48.5 - 84.0i)T^{2} \)
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

−10.87576610548851072597020127575, −9.807848401426881480451370097445, −9.390927514816777582365248808150, −8.374676037484725363138680263264, −7.47897322665751927344510586977, −6.49510026595857537164587377438, −5.50841083922447117329622799531, −3.98080213544347399036977620105, −2.96173949918806852952116399829, −1.94957397716061648211044735432, 1.27205997832606798508002195233, 2.86278155430065489505672978687, 3.93140820612461230562721842349, 4.92303354995298545495882426623, 6.37058429453415171570258475763, 7.33550921818794960357418861099, 8.130289397833142726130754078599, 9.229368867658692479804823425620, 9.680334272289337270637445972452, 10.58902546947439132122560392137

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