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Properties

Label 2-1520-19.11-c1-0-37
Degree $2$
Conductor $1520$
Sign $-0.971 - 0.235i$
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  + (1.52 − 2.63i)3-s + (−0.5 + 0.866i)5-s + 0.609·7-s + (−3.14 − 5.44i)9-s − 4.48·11-s + (−2.21 − 3.84i)13-s + (1.52 + 2.63i)15-s + (−1.45 + 2.51i)17-s + (−3.60 + 2.44i)19-s + (0.928 − 1.60i)21-s + (−1.42 − 2.46i)23-s + (−0.499 − 0.866i)25-s − 10.0·27-s + (−0.558 − 0.966i)29-s + 6.22·31-s + ⋯
L(s)  = 1  + (0.879 − 1.52i)3-s + (−0.223 + 0.387i)5-s + 0.230·7-s + (−1.04 − 1.81i)9-s − 1.35·11-s + (−0.615 − 1.06i)13-s + (0.393 + 0.681i)15-s + (−0.352 + 0.609i)17-s + (−0.827 + 0.562i)19-s + (0.202 − 0.350i)21-s + (−0.296 − 0.514i)23-s + (−0.0999 − 0.173i)25-s − 1.92·27-s + (−0.103 − 0.179i)29-s + 1.11·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.971 - 0.235i$
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: \((2,\ 1520,\ (\ :1/2),\ -0.971 - 0.235i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9955447928\)
\(L(\frac12)\) \(\approx\) \(0.9955447928\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 - 2.44i)T \)
good3 \( 1 + (-1.52 + 2.63i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 0.609T + 7T^{2} \)
11 \( 1 + 4.48T + 11T^{2} \)
13 \( 1 + (2.21 + 3.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.45 - 2.51i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.42 + 2.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.558 + 0.966i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.22T + 31T^{2} \)
37 \( 1 + 3.77T + 37T^{2} \)
41 \( 1 + (-4.15 + 7.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.99 - 8.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.94 + 5.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.22 + 7.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.11 + 8.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.49 - 4.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.23 - 7.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.80 + 10.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.86 - 3.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.51 + 7.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.12T + 83T^{2} \)
89 \( 1 + (3.96 + 6.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.83 + 8.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.485929060140432133405246738435, −8.169056033435400965594280203910, −7.64313229693958104142871155146, −6.75877697559628381508127988671, −6.04353408180393833287352503144, −4.92941285733746910279880201276, −3.50566300119912289913359451543, −2.61955652257050509348914854238, −1.93953733543096489774511647582, −0.31348889387892447092718346522, 2.20810986168350107341368007094, 2.99122302212572584689176181222, 4.15529622548916527069354240359, 4.74145685839509104731879609680, 5.31254120156239480015882168846, 6.76787580541024699059562443480, 7.88447955993662227592146076950, 8.383228261101362862587408096318, 9.249304860557456713849085007517, 9.700126296100675691647287711512

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