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Properties

Label 2-3800-760.539-c0-0-4
Degree $2$
Conductor $3800$
Sign $-0.963 - 0.269i$
Analytic cond. $1.89644$
Root an. cond. $1.37711$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−1.73 + i)3-s + (0.499 − 0.866i)4-s + (−0.999 + 1.73i)6-s − 0.999i·8-s + (1.49 − 2.59i)9-s − 11-s + 1.99i·12-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s − 3i·18-s − 19-s + (−0.866 + 0.5i)22-s + (1 + 1.73i)24-s + 4i·27-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−1.73 + i)3-s + (0.499 − 0.866i)4-s + (−0.999 + 1.73i)6-s − 0.999i·8-s + (1.49 − 2.59i)9-s − 11-s + 1.99i·12-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s − 3i·18-s − 19-s + (−0.866 + 0.5i)22-s + (1 + 1.73i)24-s + 4i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.963 - 0.269i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ -0.963 - 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01350953892\)
\(L(\frac12)\) \(\approx\) \(0.01350953892\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + 2iT - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.369131133561920805640574919438, −7.05547962657994312368423929877, −6.38409383966255564658273034159, −5.91875364966140385447326713004, −5.06065578028761016041746096033, −4.61715981751012187409223125149, −3.95294462717315068304958301786, −2.96105634288664350026406552648, −1.57257215932757804336470389349, −0.00672197456157567302788981606, 1.81120807287227144280810705385, 2.60176423012445083647290627784, 4.11791924272882553828010741323, 4.93279233357205246477165754355, 5.32957707757763564411021223001, 6.17603316675996591401192436771, 6.66349578631636608717212898733, 7.30621949582748990158951714513, 7.917537562465171470031669096354, 8.708818128720320373963947214225

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