↑

Properties

Label 2-504-56.53-c1-0-23
Degree $2$
Conductor $504$
Sign $-0.681 + 0.732i$
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  + (−0.349 − 1.37i)2-s + (−1.75 + 0.957i)4-s + (−2.26 + 1.30i)5-s + (2.62 − 0.358i)7-s + (1.92 + 2.07i)8-s + (2.58 + 2.64i)10-s + (−1.87 − 1.08i)11-s − 6.50i·13-s + (−1.40 − 3.46i)14-s + (2.16 − 3.36i)16-s + (0.797 − 1.38i)17-s + (1.27 − 0.736i)19-s + (2.72 − 4.46i)20-s + (−0.828 + 2.94i)22-s + (−1.92 − 3.33i)23-s + ⋯
L(s)  = 1  + (−0.247 − 0.969i)2-s + (−0.877 + 0.478i)4-s + (−1.01 + 0.584i)5-s + (0.990 − 0.135i)7-s + (0.680 + 0.732i)8-s + (0.816 + 0.836i)10-s + (−0.565 − 0.326i)11-s − 1.80i·13-s + (−0.376 − 0.926i)14-s + (0.541 − 0.840i)16-s + (0.193 − 0.335i)17-s + (0.292 − 0.169i)19-s + (0.608 − 0.997i)20-s + (−0.176 + 0.628i)22-s + (−0.401 − 0.695i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.681 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.681 + 0.732i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.681 + 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.330510 - 0.759005i\)
\(L(\frac12)\) \(\approx\) \(0.330510 - 0.759005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.349 + 1.37i)T \)
3 \( 1 \)
7 \( 1 + (-2.62 + 0.358i)T \)
good5 \( 1 + (2.26 - 1.30i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.87 + 1.08i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.50iT - 13T^{2} \)
17 \( 1 + (-0.797 + 1.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.27 + 0.736i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.92 + 3.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.39iT - 29T^{2} \)
31 \( 1 + (-1.79 + 3.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.69 - 3.86i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.70T + 41T^{2} \)
43 \( 1 - 2.69iT - 43T^{2} \)
47 \( 1 + (5.77 + 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.66 - 5.00i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.11 - 4.68i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.60 + 2.08i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.99 + 5.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.62 + 6.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.20iT - 83T^{2} \)
89 \( 1 + (-3.85 - 6.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.828T + 97T^{2} \)
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

−10.53165498934404054825019764167, −10.20316041752824056957190144455, −8.675207056847849259654873941072, −7.903113080761230189751766351083, −7.52598965847931188043462600853, −5.63761507619389731927583668922, −4.59677568035233622799102434836, −3.49830415834910703378783497609, −2.53441707303190712837617626732, −0.58170944483145305298450949053, 1.55743336657868097264563546475, 3.96384876613902256595893068315, 4.68031101885457045740791479193, 5.57188058845869928859469081872, 6.98237586291263748892326022909, 7.62419113765310692322217049883, 8.506032693266206624914566565210, 9.053374739732546038407707613115, 10.22419174764973281182625221493, 11.31790430777206092497057083502

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