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Properties

Label 2-1520-19.11-c1-0-0
Degree $2$
Conductor $1520$
Sign $-0.973 + 0.230i$
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  + (−0.189 + 0.328i)3-s + (−0.5 + 0.866i)5-s − 1.89·7-s + (1.42 + 2.47i)9-s − 0.134·11-s + (−1.75 − 3.04i)13-s + (−0.189 − 0.328i)15-s + (0.830 − 1.43i)17-s + (−2.10 + 3.81i)19-s + (0.359 − 0.621i)21-s + (2.68 + 4.65i)23-s + (−0.499 − 0.866i)25-s − 2.22·27-s + (−2.48 − 4.30i)29-s − 6.56·31-s + ⋯
L(s)  = 1  + (−0.109 + 0.189i)3-s + (−0.223 + 0.387i)5-s − 0.715·7-s + (0.476 + 0.824i)9-s − 0.0405·11-s + (−0.487 − 0.843i)13-s + (−0.0489 − 0.0848i)15-s + (0.201 − 0.348i)17-s + (−0.483 + 0.875i)19-s + (0.0783 − 0.135i)21-s + (0.559 + 0.969i)23-s + (−0.0999 − 0.173i)25-s − 0.427·27-s + (−0.461 − 0.799i)29-s − 1.17·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.973 + 0.230i$
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: \((2,\ 1520,\ (\ :1/2),\ -0.973 + 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2051606135\)
\(L(\frac12)\) \(\approx\) \(0.2051606135\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 - 3.81i)T \)
good3 \( 1 + (0.189 - 0.328i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 1.89T + 7T^{2} \)
11 \( 1 + 0.134T + 11T^{2} \)
13 \( 1 + (1.75 + 3.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.830 + 1.43i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.68 - 4.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.48 + 4.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.56T + 31T^{2} \)
37 \( 1 + 1.69T + 37T^{2} \)
41 \( 1 + (5.31 - 9.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.25 + 7.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.55 + 9.62i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.132 - 0.229i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.44 - 5.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.58 + 7.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.47 + 2.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.664 + 1.15i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.17 + 5.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.733 - 1.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.44T + 83T^{2} \)
89 \( 1 + (4.86 + 8.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.73 - 15.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.982061422963699771565645924511, −9.295004206791034694842500681008, −8.093333469958397038730056392703, −7.54661847795971302891303492943, −6.75577924479489807625790301007, −5.71335706095580309807124101733, −5.01855245397624295395470435755, −3.85100646275465312001260682401, −3.06911508356420823020047129651, −1.84562970725537258309089945938, 0.079465659315318079634297236963, 1.55082738815426603998228757966, 2.92881321296041894219250331758, 3.95791297018816450298980362253, 4.74306034816901942716963773820, 5.84845944077527642992754718125, 6.81265250490521818448562821953, 7.12128345077768583163870633923, 8.355209226063966526157640088860, 9.218533502601287286235935209803

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