↑

Properties

Label 2-504-7.4-c3-0-11
Degree $2$
Conductor $504$
Sign $0.386 - 0.922i$
Analytic cond. $29.7369$
Root an. cond. $5.45316$
Motivic weight $3$
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 + 1.73i)5-s + (17.5 + 6.06i)7-s + (−9 + 15.5i)11-s + 33·13-s + (−34 + 58.8i)17-s + (−12.5 − 21.6i)19-s + (46 + 79.6i)23-s + (60.5 − 104. i)25-s − 92·29-s + (−12.5 + 21.6i)31-s + (7 + 36.3i)35-s + (106.5 + 184. i)37-s − 94·41-s − 67·43-s + (139 + 240. i)47-s + ⋯
L(s)  = 1  + (0.0894 + 0.154i)5-s + (0.944 + 0.327i)7-s + (−0.246 + 0.427i)11-s + 0.704·13-s + (−0.485 + 0.840i)17-s + (−0.150 − 0.261i)19-s + (0.417 + 0.722i)23-s + (0.483 − 0.838i)25-s − 0.589·29-s + (−0.0724 + 0.125i)31-s + (0.0338 + 0.175i)35-s + (0.473 + 0.819i)37-s − 0.358·41-s − 0.237·43-s + (0.431 + 0.747i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.045141428\)
\(L(\frac12)\) \(\approx\) \(2.045141428\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + (-17.5 - 6.06i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (9 - 15.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 33T + 2.19e3T^{2} \)
17 \( 1 + (34 - 58.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (12.5 + 21.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-46 - 79.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 92T + 2.43e4T^{2} \)
31 \( 1 + (12.5 - 21.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-106.5 - 184. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 94T + 6.89e4T^{2} \)
43 \( 1 + 67T + 7.95e4T^{2} \)
47 \( 1 + (-139 - 240. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (200 - 346. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-372 + 644. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-367 - 635. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (277.5 - 480. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 642T + 3.57e5T^{2} \)
73 \( 1 + (486.5 - 842. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-392.5 - 679. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 822T + 5.71e5T^{2} \)
89 \( 1 + (-212 - 367. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 734T + 9.12e5T^{2} \)
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

−10.84448497780590900401487569501, −9.825568830339704938208297649108, −8.738524666697689042092121810682, −8.118910728632192621356487276878, −7.03948788971483805279409353981, −6.00728708527685648783256212531, −4.99645727649013367802070005141, −3.98758468412116012676776516751, −2.52039484806423175665483140606, −1.34048080309569420127970238561, 0.68580126954750621207105629629, 2.03867728755528928877148073585, 3.50224299961744865761896948536, 4.68111099028976048695459982949, 5.52892362028891005324249835016, 6.72508896054243174313639700596, 7.69247032325938594386099219158, 8.568377521143275367648541111234, 9.312483268327696272474633771507, 10.59636914930532886479695776358

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