Properties

Label 2-1520-380.379-c1-0-22
Degree $2$
Conductor $1520$
Sign $0.293 - 0.955i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.63 + 1.52i)5-s + 5.27·7-s + 3·9-s + 6.50i·11-s + 3.88i·17-s − 4.35i·19-s − 4·23-s + (0.362 − 4.98i)25-s + (−8.63 + 8.03i)35-s − 11.8·43-s + (−4.91 + 4.56i)45-s + 2.72·47-s + 20.8·49-s + (−9.91 − 10.6i)55-s + 11.2·61-s + ⋯
L(s)  = 1  + (−0.732 + 0.680i)5-s + 1.99·7-s + 9-s + 1.96i·11-s + 0.941i·17-s − 0.999i·19-s − 0.834·23-s + (0.0725 − 0.997i)25-s + (−1.45 + 1.35i)35-s − 1.80·43-s + (−0.732 + 0.680i)45-s + 0.397·47-s + 2.97·49-s + (−1.33 − 1.43i)55-s + 1.44·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.293 - 0.955i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.293 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.926419237\)
\(L(\frac12)\) \(\approx\) \(1.926419237\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.63 - 1.52i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 - 3T^{2} \)
7 \( 1 - 5.27T + 7T^{2} \)
11 \( 1 - 6.50iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 3.88iT - 17T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 - 2.72T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 16.0iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.886337823919349481154021037626, −8.583017920322716614909415929377, −7.924197326101635720799229039596, −7.26804981491160829385519288835, −6.75583999288048772738844609160, −5.22699789060008029786820967873, −4.44200796323259497350618588944, −4.04193100459705623697379320468, −2.28449554131356299831204552254, −1.55672970096238837123913419689, 0.843933468200481199120076252527, 1.79235697479113547948783657012, 3.46245100425127740208345670894, 4.30757168771991243504892240395, 5.06489940122860980211844062065, 5.79556211607949632014508615145, 7.13185191320970534664121945324, 8.044363560315186393722789867521, 8.229353475042505588992269480327, 9.052775334906463461582109981511

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