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Properties

Label 2-2760-2760.1829-c0-0-5
Degree $2$
Conductor $2760$
Sign $-0.451 + 0.892i$
Analytic cond. $1.37741$
Root an. cond. $1.17363$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (0.415 + 0.909i)6-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (−1.61 − 1.03i)11-s + (0.841 + 0.540i)12-s + (0.857 − 0.989i)13-s + (−0.142 − 0.989i)15-s + (−0.654 − 0.755i)16-s + (−0.118 − 0.258i)17-s + (−0.959 + 0.281i)18-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (0.415 + 0.909i)6-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (−1.61 − 1.03i)11-s + (0.841 + 0.540i)12-s + (0.857 − 0.989i)13-s + (−0.142 − 0.989i)15-s + (−0.654 − 0.755i)16-s + (−0.118 − 0.258i)17-s + (−0.959 + 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.451 + 0.892i$
Analytic conductor: \(1.37741\)
Root analytic conductor: \(1.17363\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2760} (1829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2760,\ (\ :0),\ -0.451 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9586518331\)
\(L(\frac12)\) \(\approx\) \(0.9586518331\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 + 0.540i)T \)
3 \( 1 + (0.142 - 0.989i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
good7 \( 1 + (0.142 - 0.989i)T^{2} \)
11 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
19 \( 1 + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \)
37 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
41 \( 1 + (-0.841 + 0.540i)T^{2} \)
43 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + 0.284T + T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
61 \( 1 + (0.959 - 0.281i)T^{2} \)
67 \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (-0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
97 \( 1 + (-0.841 + 0.540i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.607125740444639847852799593877, −8.237084031960656828514073095931, −7.15589063070508753934982463353, −6.07704334301138979020231496849, −5.42298934452873764336252797781, −4.84523456102611220960828229944, −3.72501231496923266913622025036, −3.34972494640330013204878763837, −2.53805902631693428891915745141, −0.44122354066506439007238637893, 1.79733298264595465974638071819, 2.73901179513484377369120103113, 3.82288665645150657809689817850, 4.65320320537228083727395726997, 5.37102223379474178258267877749, 6.28848978342651715332402741797, 7.00349220301547231786586817994, 7.58920093247563028126877344071, 8.263786616013076420772275652454, 8.648474321243113998127189379481

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