Properties

Label 2-1520-19.11-c1-0-17
Degree $2$
Conductor $1520$
Sign $0.923 - 0.384i$
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  + (−1.37 + 2.38i)3-s + (−0.5 + 0.866i)5-s − 4.11·7-s + (−2.30 − 3.98i)9-s + 6.09·11-s + (−1.17 − 2.02i)13-s + (−1.37 − 2.38i)15-s + (3.80 − 6.58i)17-s + (−1.46 + 4.10i)19-s + (5.67 − 9.82i)21-s + (−2.98 − 5.17i)23-s + (−0.499 − 0.866i)25-s + 4.43·27-s + (−0.969 − 1.67i)29-s + 1.43·31-s + ⋯
L(s)  = 1  + (−0.796 + 1.37i)3-s + (−0.223 + 0.387i)5-s − 1.55·7-s + (−0.767 − 1.32i)9-s + 1.83·11-s + (−0.324 − 0.562i)13-s + (−0.356 − 0.616i)15-s + (0.922 − 1.59i)17-s + (−0.335 + 0.942i)19-s + (1.23 − 2.14i)21-s + (−0.623 − 1.07i)23-s + (−0.0999 − 0.173i)25-s + 0.852·27-s + (−0.180 − 0.311i)29-s + 0.257·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.923 - 0.384i$
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: \((2,\ 1520,\ (\ :1/2),\ 0.923 - 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8733990406\)
\(L(\frac12)\) \(\approx\) \(0.8733990406\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.46 - 4.10i)T \)
good3 \( 1 + (1.37 - 2.38i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 4.11T + 7T^{2} \)
11 \( 1 - 6.09T + 11T^{2} \)
13 \( 1 + (1.17 + 2.02i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.80 + 6.58i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.98 + 5.17i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.969 + 1.67i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.43T + 31T^{2} \)
37 \( 1 - 8.91T + 37T^{2} \)
41 \( 1 + (1.70 - 2.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.655 - 1.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.192 - 0.333i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.597 + 1.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.48 + 9.50i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0825 + 0.142i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.848 + 1.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.75 + 8.23i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.97 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.87 - 6.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.25T + 83T^{2} \)
89 \( 1 + (-9.26 - 16.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.203 + 0.352i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.725690721601289625295951058463, −9.218823577978735240899100675113, −7.940872268042911297525275774099, −6.69221736456632732880894990054, −6.29457751300842831320143507649, −5.42282981420148049534537437640, −4.29185654178883232580324378185, −3.69349664702591978330151321224, −2.87411410273182219067146150301, −0.53134618559560508351567461789, 0.898947971066911799118746351854, 1.87865137327509091700824184385, 3.42444792402558826632620633184, 4.26745606766092765917564243941, 5.82591715686890142805951580062, 6.18124874678409733234503209886, 6.88955253825412154499983608254, 7.50204562442610057350000264773, 8.637545236900463259708298732464, 9.362131498308489800303168173483

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