Properties

Label 2-1520-19.11-c1-0-22
Degree $2$
Conductor $1520$
Sign $0.857 - 0.514i$
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  + (−0.232 + 0.402i)3-s + (0.5 − 0.866i)5-s + 4.83·7-s + (1.39 + 2.41i)9-s + 2.46·11-s + (2.23 + 3.87i)13-s + (0.232 + 0.402i)15-s + (−1.30 + 2.26i)17-s + (0.851 − 4.27i)19-s + (−1.12 + 1.94i)21-s + (−0.927 − 1.60i)23-s + (−0.499 − 0.866i)25-s − 2.68·27-s + (2.19 + 3.80i)29-s − 8.61·31-s + ⋯
L(s)  = 1  + (−0.134 + 0.232i)3-s + (0.223 − 0.387i)5-s + 1.82·7-s + (0.464 + 0.803i)9-s + 0.743·11-s + (0.620 + 1.07i)13-s + (0.0599 + 0.103i)15-s + (−0.317 + 0.549i)17-s + (0.195 − 0.980i)19-s + (−0.245 + 0.424i)21-s + (−0.193 − 0.334i)23-s + (−0.0999 − 0.173i)25-s − 0.517·27-s + (0.407 + 0.706i)29-s − 1.54·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.857 - 0.514i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.857 - 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.293263009\)
\(L(\frac12)\) \(\approx\) \(2.293263009\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.851 + 4.27i)T \)
good3 \( 1 + (0.232 - 0.402i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 4.83T + 7T^{2} \)
11 \( 1 - 2.46T + 11T^{2} \)
13 \( 1 + (-2.23 - 3.87i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.30 - 2.26i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.927 + 1.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.19 - 3.80i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.61T + 31T^{2} \)
37 \( 1 + 5.76T + 37T^{2} \)
41 \( 1 + (2.62 - 4.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.74 + 4.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.20 - 9.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.94 + 5.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.531 - 0.921i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.07 + 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.98 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.201 + 0.348i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.49 - 9.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.76 + 9.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 + (-3.30 - 5.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.60 + 7.98i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.252127187054374828979871049347, −8.852824354849236426873007490109, −8.002216613133921115359248662212, −7.21701982001048536865678589706, −6.25604899549924526961073341691, −5.07441853359744808843249040946, −4.68684912860796024967609606181, −3.84213069198764951103072664370, −1.99061619739466079769792184893, −1.47428249780151644717969714494, 1.10426160358656272286730948281, 1.96100422276753790288348092539, 3.47057940230439593159554132123, 4.28089132861758014683263627656, 5.41736423859037669947254840168, 6.01416624990045349968931244003, 7.14670693084497251031215848763, 7.69359239744901148290029788019, 8.586725602987388193986081037225, 9.288276124818939796786893449975

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