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Properties

Label 2-3800-760.619-c0-0-1
Degree $2$
Conductor $3800$
Sign $0.699 - 0.714i$
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 + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (0.499 + 0.866i)6-s − 0.999i·8-s − 11-s − 0.999i·12-s + (−0.5 + 0.866i)16-s + (−1.73 − i)17-s + (0.5 + 0.866i)19-s + (0.866 + 0.5i)22-s + (−0.5 + 0.866i)24-s + i·27-s + (0.866 − 0.499i)32-s + (0.866 + 0.5i)33-s + (0.999 + 1.73i)34-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (0.499 + 0.866i)6-s − 0.999i·8-s − 11-s − 0.999i·12-s + (−0.5 + 0.866i)16-s + (−1.73 − i)17-s + (0.5 + 0.866i)19-s + (0.866 + 0.5i)22-s + (−0.5 + 0.866i)24-s + i·27-s + (0.866 − 0.499i)32-s + (0.866 + 0.5i)33-s + (0.999 + 1.73i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.699 - 0.714i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (2899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.699 - 0.714i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2605323674\)
\(L(\frac12)\) \(\approx\) \(0.2605323674\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - iT - T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.999782122006383362717197497356, −7.902287453214383514613415114526, −7.44309865919673925092091514470, −6.65753264172927442631627487533, −6.00875899359188988154341987657, −5.09369246435461083925128998982, −4.12743118762662828247110578370, −2.97607261905949125944161295219, −2.21192457579688578550143415161, −0.979565346964776444200058671677, 0.26214172676901085807433587199, 1.92268295060124838718515251018, 2.82838289726690583673225602858, 4.41663682811393252382404712365, 4.90998552981309850532456330906, 5.79830443028639570780183423752, 6.27032661951362886649701181810, 7.16455374899645007595190071954, 7.86131877030068228895938708453, 8.630824164709883021520149138328

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