↑

Properties

Label 2-2925-585.34-c0-0-1
Degree $2$
Conductor $2925$
Sign $-0.504 + 0.863i$
Analytic cond. $1.45976$
Root an. cond. $1.20820$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + (0.866 − 0.5i)4-s − 9-s + (−0.366 − 1.36i)11-s + (−0.5 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.499 − 0.866i)16-s + 17-s + (1 + i)19-s + (−0.5 − 0.866i)23-s + i·27-s + (0.5 − 0.866i)29-s + (−1.36 + 0.366i)33-s + (−0.866 + 0.5i)36-s + (−1 − i)37-s + ⋯
L(s)  = 1  − i·3-s + (0.866 − 0.5i)4-s − 9-s + (−0.366 − 1.36i)11-s + (−0.5 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.499 − 0.866i)16-s + 17-s + (1 + i)19-s + (−0.5 − 0.866i)23-s + i·27-s + (0.5 − 0.866i)29-s + (−1.36 + 0.366i)33-s + (−0.866 + 0.5i)36-s + (−1 − i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.504 + 0.863i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (2374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ -0.504 + 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.360205341\)
\(L(\frac12)\) \(\approx\) \(1.360205341\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
13 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
7 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (-1 - i)T + iT^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
89 \( 1 + (1 - i)T - iT^{2} \)
97 \( 1 + (-0.866 + 0.5i)T^{2} \)
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

−8.315222063323917573243149809035, −7.946644128802348239789313131029, −7.12068747176403167873688444425, −6.47099567083235381952917459187, −5.67076695795232052900061332003, −5.29214090971954997058631704144, −3.63327893809342699536444885993, −2.77435342861800200094304102167, −1.95009505503169985153488225274, −0.820728167273032855232977630887, 1.80380979485841950187107474127, 3.00647933460644204504636035842, 3.32358608725163292942756334158, 4.74426520146407183586385938776, 5.06402006879757947657063656468, 6.12073071913905429490213162599, 7.11513337470909873168045931599, 7.64107208072722631448034000518, 8.327877111973831973535205767183, 9.445270006806308735419648163106

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