↑

Properties

Label 2-504-63.16-c1-0-17
Degree $2$
Conductor $504$
Sign $0.994 + 0.104i$
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  + (1.72 + 0.181i)3-s + 2.10·5-s + (−1.78 − 1.95i)7-s + (2.93 + 0.625i)9-s + 0.399·11-s + (1.44 + 2.49i)13-s + (3.62 + 0.381i)15-s + (−0.176 − 0.305i)17-s + (2.84 − 4.93i)19-s + (−2.71 − 3.68i)21-s − 0.877·23-s − 0.571·25-s + (4.94 + 1.60i)27-s + (0.874 − 1.51i)29-s + (−4.56 + 7.91i)31-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)3-s + 0.941·5-s + (−0.674 − 0.738i)7-s + (0.978 + 0.208i)9-s + 0.120·11-s + (0.400 + 0.693i)13-s + (0.935 + 0.0985i)15-s + (−0.0428 − 0.0741i)17-s + (0.653 − 1.13i)19-s + (−0.593 − 0.804i)21-s − 0.182·23-s − 0.114·25-s + (0.950 + 0.309i)27-s + (0.162 − 0.281i)29-s + (−0.820 + 1.42i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.994 + 0.104i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.994 + 0.104i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20743 - 0.115165i\)
\(L(\frac12)\) \(\approx\) \(2.20743 - 0.115165i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.181i)T \)
7 \( 1 + (1.78 + 1.95i)T \)
good5 \( 1 - 2.10T + 5T^{2} \)
11 \( 1 - 0.399T + 11T^{2} \)
13 \( 1 + (-1.44 - 2.49i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.176 + 0.305i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.84 + 4.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.877T + 23T^{2} \)
29 \( 1 + (-0.874 + 1.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.56 - 7.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.39 - 5.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.20 - 2.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.276 + 0.479i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.86 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.07 + 3.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.66 + 8.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.03 - 8.72i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.601 - 1.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + (-0.315 - 0.546i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.24 - 2.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.59 - 7.95i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.29 + 12.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.84 - 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.59364313581740396226418851353, −9.842602270186359458887246126602, −9.262537771476023361381414755058, −8.404976457794934501149056549245, −7.11177015660119844487028039779, −6.59494777955240833801046789825, −5.15353846632121513416255788334, −3.93974692621782328869453407779, −2.91727013025156978911375337054, −1.57562854729862564496882429416, 1.73851843158847415459139872757, 2.85173688046901277572882723468, 3.87278047328155861178029353761, 5.55265237679813410663185079729, 6.18627281057966197067447271898, 7.44459187551612407360104401112, 8.342726729612328186710607236677, 9.333171723822275546739914549405, 9.728052661974204243433894626917, 10.66404212400800237259833932234

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