↑

Properties

Label 2-504-56.37-c1-0-4
Degree $2$
Conductor $504$
Sign $-0.00767 - 0.999i$
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  + (−0.902 + 1.08i)2-s + (−0.371 − 1.96i)4-s + (−3.08 − 1.78i)5-s + (−2.38 + 1.14i)7-s + (2.47 + 1.36i)8-s + (4.72 − 1.75i)10-s + (3.52 − 2.03i)11-s + 1.44i·13-s + (0.899 − 3.63i)14-s + (−3.72 + 1.46i)16-s + (3.49 + 6.05i)17-s + (−0.261 − 0.150i)19-s + (−2.35 + 6.73i)20-s + (−0.964 + 5.67i)22-s + (−1.21 + 2.10i)23-s + ⋯
L(s)  = 1  + (−0.637 + 0.770i)2-s + (−0.185 − 0.982i)4-s + (−1.38 − 0.797i)5-s + (−0.900 + 0.434i)7-s + (0.875 + 0.483i)8-s + (1.49 − 0.554i)10-s + (1.06 − 0.613i)11-s + 0.399i·13-s + (0.240 − 0.970i)14-s + (−0.930 + 0.365i)16-s + (0.847 + 1.46i)17-s + (−0.0599 − 0.0345i)19-s + (−0.526 + 1.50i)20-s + (−0.205 + 1.21i)22-s + (−0.252 + 0.437i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00767 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00767 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.00767 - 0.999i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.00767 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.430079 + 0.433394i\)
\(L(\frac12)\) \(\approx\) \(0.430079 + 0.433394i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.902 - 1.08i)T \)
3 \( 1 \)
7 \( 1 + (2.38 - 1.14i)T \)
good5 \( 1 + (3.08 + 1.78i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.52 + 2.03i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.44iT - 13T^{2} \)
17 \( 1 + (-3.49 - 6.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.261 + 0.150i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.21 - 2.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.151iT - 29T^{2} \)
31 \( 1 + (-2.37 - 4.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.82 - 5.67i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.239T + 41T^{2} \)
43 \( 1 + 1.32iT - 43T^{2} \)
47 \( 1 + (3.17 - 5.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.18 + 2.99i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.73 - 5.62i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.64 + 2.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.79 - 2.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 + (0.284 + 0.493i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.746 + 1.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.0iT - 83T^{2} \)
89 \( 1 + (1.83 - 3.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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

−11.15321632703390227333041802900, −9.972134125380621060979286252672, −9.096342635765799854200957363097, −8.440931951595637175471949331475, −7.74342473106600083562664367413, −6.57013028591844096475524595122, −5.84654646087589305014207806020, −4.48285165674980451972925864356, −3.53628060884944220377937163405, −1.14989069572428032002681661608, 0.56061280251399636637470603061, 2.75493642706107853285408503430, 3.62234728868692809804552362363, 4.41881070542506790080664685238, 6.48084773185502781448436952868, 7.37118909756634818641065442286, 7.82608972949661590198231008843, 9.181913294874055502663702043298, 9.855828248913505776181781783279, 10.70772459931340396516387736324

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