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Properties

Label 2-3800-152.83-c0-0-2
Degree $2$
Conductor $3800$
Sign $-0.671 + 0.740i$
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  + (0.5 + 0.866i)2-s + (1 + 1.73i)3-s + (−0.499 + 0.866i)4-s + (−0.999 + 1.73i)6-s − 0.999·8-s + (−1.49 + 2.59i)9-s − 11-s − 1.99·12-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s − 3·18-s + 19-s + (−0.5 − 0.866i)22-s + (−1 − 1.73i)24-s − 4·27-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (1 + 1.73i)3-s + (−0.499 + 0.866i)4-s + (−0.999 + 1.73i)6-s − 0.999·8-s + (−1.49 + 2.59i)9-s − 11-s − 1.99·12-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s − 3·18-s + 19-s + (−0.5 − 0.866i)22-s + (−1 − 1.73i)24-s − 4·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.671 + 0.740i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ -0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.753540402\)
\(L(\frac12)\) \(\approx\) \(1.753540402\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.114906605203099348350691795048, −8.382015224424303578515150486612, −7.79762057789068554485671239146, −7.16848059185293088992999408260, −5.78220173039908980144772048898, −5.28638264262488928653287288893, −4.54594366836949936912247901336, −4.03281896099082884616828639694, −2.87706574796491818910088509642, −2.72319435301908874115882307727, 0.71116573236821241728007108736, 1.86938424870237193609413054733, 2.42421003802135871988035216899, 3.24833178938098261104631913211, 3.98462332578615772159293043261, 5.35735827146484612942498489823, 5.95113986075958895750743345774, 6.81476535246409485699317389686, 7.52537154744642913452819390419, 8.214388630660322052894054126541

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