Properties

Label 2-5520-92.91-c1-0-64
Degree $2$
Conductor $5520$
Sign $-0.226 + 0.974i$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
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  i·3-s + i·5-s + 0.279·7-s − 9-s + 0.628·11-s − 3.03·13-s + 15-s + 2.36i·17-s − 1.48·19-s − 0.279i·21-s + (−1.08 + 4.67i)23-s − 25-s + i·27-s + 3.72·29-s − 8.99i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.447i·5-s + 0.105·7-s − 0.333·9-s + 0.189·11-s − 0.840·13-s + 0.258·15-s + 0.573i·17-s − 0.341·19-s − 0.0609i·21-s + (−0.226 + 0.974i)23-s − 0.200·25-s + 0.192i·27-s + 0.691·29-s − 1.61i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.226 + 0.974i$
Analytic conductor: \(44.0774\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ -0.226 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.182995506\)
\(L(\frac12)\) \(\approx\) \(1.182995506\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 - iT \)
23 \( 1 + (1.08 - 4.67i)T \)
good7 \( 1 - 0.279T + 7T^{2} \)
11 \( 1 - 0.628T + 11T^{2} \)
13 \( 1 + 3.03T + 13T^{2} \)
17 \( 1 - 2.36iT - 17T^{2} \)
19 \( 1 + 1.48T + 19T^{2} \)
29 \( 1 - 3.72T + 29T^{2} \)
31 \( 1 + 8.99iT - 31T^{2} \)
37 \( 1 + 5.99iT - 37T^{2} \)
41 \( 1 + 7.39T + 41T^{2} \)
43 \( 1 - 8.69T + 43T^{2} \)
47 \( 1 + 2.41iT - 47T^{2} \)
53 \( 1 - 1.22iT - 53T^{2} \)
59 \( 1 + 7.12iT - 59T^{2} \)
61 \( 1 + 3.34iT - 61T^{2} \)
67 \( 1 + 8.37T + 67T^{2} \)
71 \( 1 - 3.78iT - 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 - 16.5T + 79T^{2} \)
83 \( 1 - 7.07T + 83T^{2} \)
89 \( 1 + 12.8iT - 89T^{2} \)
97 \( 1 + 12.7iT - 97T^{2} \)
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

−7.82940845492149002780051065598, −7.31216019013528945643421631377, −6.53360011229036435782522311669, −5.94359448382427117468718823878, −5.14242938122155297247203683648, −4.19823977985987989328623097761, −3.39610162143559747776651149202, −2.40278291185133705788836485522, −1.73436854261018731109978094622, −0.34029791827950984980731671926, 1.00527118003519328107305890906, 2.27072993205480713990149929763, 3.08187285983055178170204415302, 4.03820085006013574815825021261, 4.92001895677526138486468600797, 5.07468951683662584162381917500, 6.30293963490573583569682244945, 6.81272581384217756393600563764, 7.81067114589855464706255941106, 8.389253956975886961742437054840

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