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Properties

Label 2-1520-19.11-c1-0-19
Degree $2$
Conductor $1520$
Sign $0.998 - 0.0531i$
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  + (0.595 − 1.03i)3-s + (−0.5 + 0.866i)5-s + 3.62·7-s + (0.790 + 1.36i)9-s + 1.46·11-s + (−0.828 − 1.43i)13-s + (0.595 + 1.03i)15-s + (0.709 − 1.22i)17-s + (−2.75 + 3.38i)19-s + (2.16 − 3.74i)21-s + (3.55 + 6.16i)23-s + (−0.499 − 0.866i)25-s + 5.45·27-s + (−0.704 − 1.22i)29-s + 2.45·31-s + ⋯
L(s)  = 1  + (0.343 − 0.595i)3-s + (−0.223 + 0.387i)5-s + 1.37·7-s + (0.263 + 0.456i)9-s + 0.441·11-s + (−0.229 − 0.397i)13-s + (0.153 + 0.266i)15-s + (0.171 − 0.297i)17-s + (−0.631 + 0.775i)19-s + (0.471 − 0.816i)21-s + (0.741 + 1.28i)23-s + (−0.0999 − 0.173i)25-s + 1.05·27-s + (−0.130 − 0.226i)29-s + 0.441·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.998 - 0.0531i$
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: \((2,\ 1520,\ (\ :1/2),\ 0.998 - 0.0531i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.276185756\)
\(L(\frac12)\) \(\approx\) \(2.276185756\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.75 - 3.38i)T \)
good3 \( 1 + (-0.595 + 1.03i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 3.62T + 7T^{2} \)
11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 + (0.828 + 1.43i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.709 + 1.22i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.55 - 6.16i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.704 + 1.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.45T + 31T^{2} \)
37 \( 1 - 0.865T + 37T^{2} \)
41 \( 1 + (1.62 - 2.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.276 + 0.478i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.67 - 2.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.53 + 6.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.67 - 4.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.34 + 5.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.39 - 2.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.50 + 7.80i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.53 - 4.39i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.50 + 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + (-5.84 - 10.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.44 + 9.42i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.405016819285104582026929385332, −8.382915843013076745582902168982, −7.82718821745606259247635471979, −7.34151789957637204149559018814, −6.34891326075618223873991445227, −5.24327179666880636021432366655, −4.50560966434648877013672142065, −3.35149446918014752896624418764, −2.15846382983342810647712414375, −1.31130260943764113955093351201, 1.04711080435630049910548461609, 2.31074033869788008170003705551, 3.63383054549466203569917912919, 4.58391552778608978742724058909, 4.85508556626251390936679590601, 6.25058604387071473736308310818, 7.10089788981487380899714160653, 8.062130632898371870781275513035, 8.816896579399066924472574705745, 9.180852283828235799535950105798

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