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Properties

Label 2-3789-421.203-c0-0-0
Degree $2$
Conductor $3789$
Sign $-0.778 + 0.627i$
Analytic cond. $1.89095$
Root an. cond. $1.37512$
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  + (−0.722 − 0.691i)4-s + (−0.355 − 1.95i)7-s + (0.746 − 1.46i)13-s + (0.0448 + 0.998i)16-s + (1.52 − 1.27i)19-s + (−0.0448 + 0.998i)25-s + (−1.09 + 1.66i)28-s + (−0.258 − 0.0714i)31-s + (0.457 + 1.52i)37-s + (−1.39 − 1.27i)43-s + (−2.77 + 1.04i)49-s + (−1.55 + 0.542i)52-s + (0.388 − 0.424i)61-s + (0.657 − 0.753i)64-s + (−0.975 − 0.316i)67-s + ⋯
L(s)  = 1  + (−0.722 − 0.691i)4-s + (−0.355 − 1.95i)7-s + (0.746 − 1.46i)13-s + (0.0448 + 0.998i)16-s + (1.52 − 1.27i)19-s + (−0.0448 + 0.998i)25-s + (−1.09 + 1.66i)28-s + (−0.258 − 0.0714i)31-s + (0.457 + 1.52i)37-s + (−1.39 − 1.27i)43-s + (−2.77 + 1.04i)49-s + (−1.55 + 0.542i)52-s + (0.388 − 0.424i)61-s + (0.657 − 0.753i)64-s + (−0.975 − 0.316i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3789\)    =    \(3^{2} \cdot 421\)
Sign: $-0.778 + 0.627i$
Analytic conductor: \(1.89095\)
Root analytic conductor: \(1.37512\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3789} (1045, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3789,\ (\ :0),\ -0.778 + 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9616554292\)
\(L(\frac12)\) \(\approx\) \(0.9616554292\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
421 \( 1 + (0.178 - 0.983i)T \)
good2 \( 1 + (0.722 + 0.691i)T^{2} \)
5 \( 1 + (0.0448 - 0.998i)T^{2} \)
7 \( 1 + (0.355 + 1.95i)T + (-0.936 + 0.351i)T^{2} \)
11 \( 1 + (-0.550 - 0.834i)T^{2} \)
13 \( 1 + (-0.746 + 1.46i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.134 - 0.990i)T^{2} \)
19 \( 1 + (-1.52 + 1.27i)T + (0.178 - 0.983i)T^{2} \)
23 \( 1 + (-0.266 - 0.963i)T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + (0.258 + 0.0714i)T + (0.858 + 0.512i)T^{2} \)
37 \( 1 + (-0.457 - 1.52i)T + (-0.834 + 0.550i)T^{2} \)
41 \( 1 + (-0.178 + 0.983i)T^{2} \)
43 \( 1 + (1.39 + 1.27i)T + (0.0896 + 0.995i)T^{2} \)
47 \( 1 + (0.998 + 0.0448i)T^{2} \)
53 \( 1 + (-0.351 - 0.936i)T^{2} \)
59 \( 1 + (-0.880 + 0.473i)T^{2} \)
61 \( 1 + (-0.388 + 0.424i)T + (-0.0896 - 0.995i)T^{2} \)
67 \( 1 + (0.975 + 0.316i)T + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (-0.880 - 0.473i)T^{2} \)
73 \( 1 + (-0.309 + 1.95i)T + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.0825 - 1.83i)T + (-0.995 - 0.0896i)T^{2} \)
83 \( 1 + (0.880 - 0.473i)T^{2} \)
89 \( 1 + (-0.512 - 0.858i)T^{2} \)
97 \( 1 + (-1.66 + 0.149i)T + (0.983 - 0.178i)T^{2} \)
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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

−8.348619418446582485241223731848, −7.61201358326712802643062385713, −6.98837496219713656392907041352, −6.17252287801222754967135481107, −5.21527590632534211445248470474, −4.72706947699906805589656051197, −3.61629248952938452758722496375, −3.24171907938219960164138653140, −1.32248538589638058616832914747, −0.63174193718158487510307216837, 1.70190855302774529276546631775, 2.73198480671873600977282291513, 3.52018883641913019784930150442, 4.35033051586878636740704868592, 5.31625772611579207274774433280, 5.90265382305155030323948244554, 6.66226959272773452766409331992, 7.71977650897158332482714361700, 8.377717230241079053090303384572, 8.977871092975540880213870462931

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