↑

Properties

Label 2-504-63.58-c1-0-15
Degree $2$
Conductor $504$
Sign $0.281 + 0.959i$
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  + (−1.13 + 1.30i)3-s + (−0.170 − 0.294i)5-s + (−2.63 − 0.253i)7-s + (−0.429 − 2.96i)9-s + (0.335 − 0.581i)11-s + (1.62 − 2.81i)13-s + (0.578 + 0.111i)15-s + (−1.10 − 1.90i)17-s + (0.242 − 0.419i)19-s + (3.31 − 3.16i)21-s + (−2.09 − 3.62i)23-s + (2.44 − 4.22i)25-s + (4.37 + 2.80i)27-s + (0.478 + 0.829i)29-s + 2.08·31-s + ⋯
L(s)  = 1  + (−0.654 + 0.756i)3-s + (−0.0760 − 0.131i)5-s + (−0.995 − 0.0957i)7-s + (−0.143 − 0.989i)9-s + (0.101 − 0.175i)11-s + (0.450 − 0.779i)13-s + (0.149 + 0.0287i)15-s + (−0.266 − 0.462i)17-s + (0.0555 − 0.0961i)19-s + (0.723 − 0.689i)21-s + (−0.436 − 0.756i)23-s + (0.488 − 0.845i)25-s + (0.842 + 0.539i)27-s + (0.0889 + 0.154i)29-s + 0.374·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.281 + 0.959i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.555742 - 0.415948i\)
\(L(\frac12)\) \(\approx\) \(0.555742 - 0.415948i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.13 - 1.30i)T \)
7 \( 1 + (2.63 + 0.253i)T \)
good5 \( 1 + (0.170 + 0.294i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.335 + 0.581i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.62 + 2.81i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.10 + 1.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.242 + 0.419i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.09 + 3.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.478 - 0.829i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.08T + 31T^{2} \)
37 \( 1 + (-4.81 + 8.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.90 - 6.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.66 + 6.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.69T + 47T^{2} \)
53 \( 1 + (6.12 + 10.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.94T + 59T^{2} \)
61 \( 1 + 3.52T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + 5.57T + 71T^{2} \)
73 \( 1 + (3.71 + 6.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 + (-2.47 - 4.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.52 - 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.23 - 7.33i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.62181203100503724388413550927, −9.983467583968004483815993373104, −9.144420683264554270481771295692, −8.200329053871328849282100211091, −6.76222514260708573157146240268, −6.11220028255455525357925743407, −5.04953717797818052663896949751, −3.96873773961246440872280632550, −2.93747549411360065817990550745, −0.46007265901027172541867705272, 1.55936849848962898069197405243, 3.09462423829663733921086495352, 4.46677298225509016437527807860, 5.80102044832045351247223097680, 6.49515617483772852531315295859, 7.24607903965785418778670231134, 8.327935336648837768571894600728, 9.388600667460125133247995327500, 10.28807929746169875133129315531, 11.28418158423953417428418479887

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