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Properties

Label 2-3789-421.129-c0-0-0
Degree $2$
Conductor $3789$
Sign $-0.0659 - 0.997i$
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  + (−0.178 − 0.983i)4-s + (−1.49 − 0.202i)7-s + (−1.19 + 0.189i)13-s + (−0.936 + 0.351i)16-s + (0.0150 + 0.223i)19-s + (0.936 + 0.351i)25-s + (0.0675 + 1.50i)28-s + (−0.522 + 0.790i)31-s + (−0.0429 + 1.91i)37-s + (0.601 + 0.273i)43-s + (1.22 + 0.337i)49-s + (0.400 + 1.14i)52-s + (−0.0185 + 0.0408i)61-s + (0.512 + 0.858i)64-s + (−1.08 − 1.48i)67-s + ⋯
L(s)  = 1  + (−0.178 − 0.983i)4-s + (−1.49 − 0.202i)7-s + (−1.19 + 0.189i)13-s + (−0.936 + 0.351i)16-s + (0.0150 + 0.223i)19-s + (0.936 + 0.351i)25-s + (0.0675 + 1.50i)28-s + (−0.522 + 0.790i)31-s + (−0.0429 + 1.91i)37-s + (0.601 + 0.273i)43-s + (1.22 + 0.337i)49-s + (0.400 + 1.14i)52-s + (−0.0185 + 0.0408i)61-s + (0.512 + 0.858i)64-s + (−1.08 − 1.48i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0659 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0659 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3789\)    =    \(3^{2} \cdot 421\)
Sign: $-0.0659 - 0.997i$
Analytic conductor: \(1.89095\)
Root analytic conductor: \(1.37512\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3789} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3789,\ (\ :0),\ -0.0659 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3501714638\)
\(L(\frac12)\) \(\approx\) \(0.3501714638\)
\(L(1)\) 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 \( 1 + (-0.990 + 0.134i)T \)
good2 \( 1 + (0.178 + 0.983i)T^{2} \)
5 \( 1 + (-0.936 - 0.351i)T^{2} \)
7 \( 1 + (1.49 + 0.202i)T + (0.963 + 0.266i)T^{2} \)
11 \( 1 + (-0.0448 + 0.998i)T^{2} \)
13 \( 1 + (1.19 - 0.189i)T + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (-0.473 + 0.880i)T^{2} \)
19 \( 1 + (-0.0150 - 0.223i)T + (-0.990 + 0.134i)T^{2} \)
23 \( 1 + (-0.834 + 0.550i)T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + (0.522 - 0.790i)T + (-0.393 - 0.919i)T^{2} \)
37 \( 1 + (0.0429 - 1.91i)T + (-0.998 - 0.0448i)T^{2} \)
41 \( 1 + (0.990 - 0.134i)T^{2} \)
43 \( 1 + (-0.601 - 0.273i)T + (0.657 + 0.753i)T^{2} \)
47 \( 1 + (-0.351 + 0.936i)T^{2} \)
53 \( 1 + (0.266 - 0.963i)T^{2} \)
59 \( 1 + (0.722 + 0.691i)T^{2} \)
61 \( 1 + (0.0185 - 0.0408i)T + (-0.657 - 0.753i)T^{2} \)
67 \( 1 + (1.08 + 1.48i)T + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.722 - 0.691i)T^{2} \)
73 \( 1 + (0.809 - 1.58i)T + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.167 - 0.0629i)T + (0.753 + 0.657i)T^{2} \)
83 \( 1 + (-0.722 - 0.691i)T^{2} \)
89 \( 1 + (-0.919 - 0.393i)T^{2} \)
97 \( 1 + (1.50 - 1.31i)T + (0.134 - 0.990i)T^{2} \)
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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

−9.277220999852832242390330824471, −8.196077968935492251916014041735, −7.06653702897437881515490058442, −6.73218234628718618399974740452, −5.94858802887762147753330820413, −5.13737866993282200915932856756, −4.44417243070774482814552964777, −3.34164683080252847114956062655, −2.55857373881326343697685405007, −1.26921412597914354489849921589, 0.20293887776635599388891736123, 2.36576682134811222116925569078, 2.92666408945124992921097687709, 3.76686955866285826392881312768, 4.54513761449929683218218867382, 5.54452557699606704293912330478, 6.36052561569752242754252877012, 7.26727000628971853873525778441, 7.47375649774842665194775975897, 8.648419444171555456599724419716

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