↑

Properties

Label 2-2775-555.359-c0-0-0
Degree $2$
Conductor $2775$
Sign $-0.667 + 0.744i$
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 + 0.5i)13-s + (−0.499 − 0.866i)16-s + (1 − 1.73i)19-s + (0.499 + 0.866i)21-s − 0.999i·27-s + (−0.866 + 0.499i)28-s − 31-s + 0.999·36-s − i·37-s + (−0.499 − 0.866i)39-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 + 0.5i)13-s + (−0.499 − 0.866i)16-s + (1 − 1.73i)19-s + (0.499 + 0.866i)21-s − 0.999i·27-s + (−0.866 + 0.499i)28-s − 31-s + 0.999·36-s − i·37-s + (−0.499 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2775\)    =    \(3 \cdot 5^{2} \cdot 37\)
Sign: $-0.667 + 0.744i$
Analytic conductor: \(1.38490\)
Root analytic conductor: \(1.17682\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2775} (2024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2775,\ (\ :0),\ -0.667 + 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8375952647\)
\(L(\frac12)\) \(\approx\) \(0.8375952647\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - iT - T^{2} \)
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

−8.969824525440690254314426691571, −7.53106042385812524248667701537, −7.02977190361615860901508633606, −6.48798319002898440444146479794, −5.74020949622263983658758002605, −5.08224361122067076829324263509, −4.04966433195771582835647299868, −2.83792109104216925563622807242, −1.68563572102360427890900640987, −0.61142252351547762562900073435, 1.55249092406693099271471259034, 3.17714460936252941051868439030, 3.46283218285290825074213414428, 4.50758924548544409562797433188, 5.73798197336295451693772336730, 6.03535144523682538884445128663, 6.88406732412464408484725196955, 7.73814356975619601084487868833, 8.483093712648298024982623883269, 9.392013133778008922310279283906

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