↑

Properties

Label 2-2775-555.344-c0-0-0
Degree $2$
Conductor $2775$
Sign $-0.769 - 0.638i$
Analytic cond. $1.38490$
Root an. cond. $1.17682$
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  + (0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)12-s + (−0.866 + 1.5i)13-s + (−0.499 − 0.866i)16-s + (−0.499 − 0.866i)21-s + 0.999i·27-s + (0.866 − 0.499i)28-s + 1.73i·31-s − 0.999·36-s + i·37-s + (−1.5 + 0.866i)39-s − 1.73·43-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)12-s + (−0.866 + 1.5i)13-s + (−0.499 − 0.866i)16-s + (−0.499 − 0.866i)21-s + 0.999i·27-s + (0.866 − 0.499i)28-s + 1.73i·31-s − 0.999·36-s + i·37-s + (−1.5 + 0.866i)39-s − 1.73·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2775\)    =    \(3 \cdot 5^{2} \cdot 37\)
Sign: $-0.769 - 0.638i$
Analytic conductor: \(1.38490\)
Root analytic conductor: \(1.17682\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2775} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2775,\ (\ :0),\ -0.769 - 0.638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9970403614\)
\(L(\frac12)\) \(\approx\) \(0.9970403614\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
37 \( 1 - iT \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + 1.73T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.73T + T^{2} \)
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

−9.204701853059893237358321185755, −8.674311308451422482992388388874, −7.87921110668924798582396507562, −7.05928996303693308676517763204, −6.62166500434896194511028762739, −4.96105797523618330106837590952, −4.52311464367830147949812882282, −3.55876433443509702611320613231, −3.08924180433997133229528868416, −1.92718824442053268999928333034, 0.54878027015358715741982972381, 2.02348480493647955834351647462, 2.87266190732957214268904877643, 3.75135460286382860063789212531, 4.85326081106356257485626276436, 5.75477604392080847559428312059, 6.30658264706871980479715539672, 7.28965200022887898801645671932, 7.995890017133682802889822601042, 8.774185336426344890259972030387

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