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Properties

Label 2-3789-421.360-c0-0-0
Degree $2$
Conductor $3789$
Sign $0.999 + 0.0147i$
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  + (0.834 + 0.550i)4-s + (1.38 − 0.0620i)7-s + (−0.670 − 1.31i)13-s + (0.393 + 0.919i)16-s + (0.0379 − 1.69i)19-s + (−0.393 + 0.919i)25-s + (1.18 + 0.708i)28-s + (1.41 − 1.23i)31-s + (0.719 + 0.408i)37-s + (0.0833 − 0.207i)43-s + (0.906 − 0.0815i)49-s + (0.165 − 1.46i)52-s + (−1.61 − 0.647i)61-s + (−0.178 + 0.983i)64-s + (−1.88 + 0.612i)67-s + ⋯
L(s)  = 1  + (0.834 + 0.550i)4-s + (1.38 − 0.0620i)7-s + (−0.670 − 1.31i)13-s + (0.393 + 0.919i)16-s + (0.0379 − 1.69i)19-s + (−0.393 + 0.919i)25-s + (1.18 + 0.708i)28-s + (1.41 − 1.23i)31-s + (0.719 + 0.408i)37-s + (0.0833 − 0.207i)43-s + (0.906 − 0.0815i)49-s + (0.165 − 1.46i)52-s + (−1.61 − 0.647i)61-s + (−0.178 + 0.983i)64-s + (−1.88 + 0.612i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3789\)    =    \(3^{2} \cdot 421\)
Sign: $0.999 + 0.0147i$
Analytic conductor: \(1.89095\)
Root analytic conductor: \(1.37512\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3789} (2044, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3789,\ (\ :0),\ 0.999 + 0.0147i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.837358247\)
\(L(\frac12)\) \(\approx\) \(1.837358247\)
\(L(1)\) 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 \( 1 + (-0.998 - 0.0448i)T \)
good2 \( 1 + (-0.834 - 0.550i)T^{2} \)
5 \( 1 + (0.393 - 0.919i)T^{2} \)
7 \( 1 + (-1.38 + 0.0620i)T + (0.995 - 0.0896i)T^{2} \)
11 \( 1 + (0.858 - 0.512i)T^{2} \)
13 \( 1 + (0.670 + 1.31i)T + (-0.587 + 0.809i)T^{2} \)
17 \( 1 + (-0.936 - 0.351i)T^{2} \)
19 \( 1 + (-0.0379 + 1.69i)T + (-0.998 - 0.0448i)T^{2} \)
23 \( 1 + (0.657 - 0.753i)T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + (-1.41 + 1.23i)T + (0.134 - 0.990i)T^{2} \)
37 \( 1 + (-0.719 - 0.408i)T + (0.512 + 0.858i)T^{2} \)
41 \( 1 + (0.998 + 0.0448i)T^{2} \)
43 \( 1 + (-0.0833 + 0.207i)T + (-0.722 - 0.691i)T^{2} \)
47 \( 1 + (-0.919 - 0.393i)T^{2} \)
53 \( 1 + (-0.0896 - 0.995i)T^{2} \)
59 \( 1 + (-0.266 + 0.963i)T^{2} \)
61 \( 1 + (1.61 + 0.647i)T + (0.722 + 0.691i)T^{2} \)
67 \( 1 + (1.88 - 0.612i)T + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (-0.266 - 0.963i)T^{2} \)
73 \( 1 + (-0.309 - 1.95i)T + (-0.951 + 0.309i)T^{2} \)
79 \( 1 + (0.692 - 1.61i)T + (-0.691 - 0.722i)T^{2} \)
83 \( 1 + (0.266 - 0.963i)T^{2} \)
89 \( 1 + (-0.990 + 0.134i)T^{2} \)
97 \( 1 + (0.708 - 0.741i)T + (-0.0448 - 0.998i)T^{2} \)
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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

−8.418507737046467483126432483068, −7.80800605107416097909932974075, −7.46218122070666067214482788558, −6.57915648390708908482144216889, −5.64289765677765476709807557414, −4.91233945031216839397913844339, −4.14818637514048551095270986754, −2.91266107176790909828340726254, −2.42208309814632657532538936897, −1.18965004859437382341332387188, 1.47801486862859037574261697236, 1.94569045846748428122245788528, 3.01247850483976314684908381699, 4.38544117127688752904200086340, 4.78098972759516798945288620990, 5.86827972299250775962226193639, 6.33948291141313698732178562645, 7.33269243072329265955945344350, 7.81738801777621760685794025616, 8.579201430356950778244934094645

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