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Properties

Label 2-1520-19.11-c1-0-16
Degree $2$
Conductor $1520$
Sign $-0.468 - 0.883i$
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  + (−1.72 + 2.98i)3-s + (−0.5 + 0.866i)5-s + 4.44·7-s + (−4.44 − 7.70i)9-s + 0.918·11-s + (1.76 + 3.05i)13-s + (−1.72 − 2.98i)15-s + (1.94 − 3.37i)17-s + (4.22 + 1.07i)19-s + (−7.67 + 13.2i)21-s + (3.44 + 5.96i)23-s + (−0.499 − 0.866i)25-s + 20.3·27-s + (0.0407 + 0.0705i)29-s + 3.00·31-s + ⋯
L(s)  = 1  + (−0.995 + 1.72i)3-s + (−0.223 + 0.387i)5-s + 1.68·7-s + (−1.48 − 2.56i)9-s + 0.276·11-s + (0.489 + 0.847i)13-s + (−0.445 − 0.771i)15-s + (0.471 − 0.817i)17-s + (0.969 + 0.246i)19-s + (−1.67 + 2.89i)21-s + (0.718 + 1.24i)23-s + (−0.0999 − 0.173i)25-s + 3.91·27-s + (0.00756 + 0.0130i)29-s + 0.539·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.468 - 0.883i$
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: \((2,\ 1520,\ (\ :1/2),\ -0.468 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.490093789\)
\(L(\frac12)\) \(\approx\) \(1.490093789\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.22 - 1.07i)T \)
good3 \( 1 + (1.72 - 2.98i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 4.44T + 7T^{2} \)
11 \( 1 - 0.918T + 11T^{2} \)
13 \( 1 + (-1.76 - 3.05i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.94 + 3.37i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.44 - 5.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0407 - 0.0705i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.00T + 31T^{2} \)
37 \( 1 - 6.44T + 37T^{2} \)
41 \( 1 + (-0.224 + 0.388i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.90 - 3.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.40 + 4.17i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.76 - 4.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.04 - 1.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.20 + 9.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.44 + 7.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.46 - 2.53i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.07 + 13.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.44 - 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.16T + 83T^{2} \)
89 \( 1 + (-1.26 - 2.18i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.27 - 5.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.666879525238190338816108542012, −9.274031333468896978621901208125, −8.273918201093456107811261505403, −7.30079588686125039787756966054, −6.20561883685209392942642719434, −5.31918911559413269589635799324, −4.79423128275420728677456428947, −4.02341980433619974835293590638, −3.12226543625697350024152140476, −1.19902684311727707524980299997, 0.896903678867556048676146006358, 1.45765420096126472702759311564, 2.68851462583963321207405743097, 4.48804837410524372347683544696, 5.28943070064734187169733556422, 5.87275267910213832229300784963, 6.84406421764902513580226992252, 7.69722959471659468340102327342, 8.134222613611434701314198220644, 8.710139336237879011178987136678

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