Properties

Label 2-1872-1.1-c3-0-43
Degree $2$
Conductor $1872$
Sign $-1$
Analytic cond. $110.451$
Root an. cond. $10.5095$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 19.3·5-s − 4.84·7-s − 61.0·11-s + 13·13-s + 41.7·17-s + 107.·19-s + 28.5·23-s + 249.·25-s + 89.8·29-s − 183.·31-s + 93.6·35-s + 418.·37-s + 142.·41-s + 71.0·43-s + 323.·47-s − 319.·49-s + 25.1·53-s + 1.18e3·55-s − 684.·59-s + 308.·61-s − 251.·65-s − 672.·67-s − 326.·71-s + 24.3·73-s + 295.·77-s − 166.·79-s − 201.·83-s + ⋯
L(s)  = 1  − 1.72·5-s − 0.261·7-s − 1.67·11-s + 0.277·13-s + 0.596·17-s + 1.29·19-s + 0.258·23-s + 1.99·25-s + 0.575·29-s − 1.06·31-s + 0.452·35-s + 1.85·37-s + 0.543·41-s + 0.252·43-s + 1.00·47-s − 0.931·49-s + 0.0650·53-s + 2.89·55-s − 1.51·59-s + 0.646·61-s − 0.479·65-s − 1.22·67-s − 0.546·71-s + 0.0389·73-s + 0.437·77-s − 0.237·79-s − 0.265·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(110.451\)
Root analytic conductor: \(10.5095\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1872,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 - 13T \)
good5 \( 1 + 19.3T + 125T^{2} \)
7 \( 1 + 4.84T + 343T^{2} \)
11 \( 1 + 61.0T + 1.33e3T^{2} \)
17 \( 1 - 41.7T + 4.91e3T^{2} \)
19 \( 1 - 107.T + 6.85e3T^{2} \)
23 \( 1 - 28.5T + 1.21e4T^{2} \)
29 \( 1 - 89.8T + 2.43e4T^{2} \)
31 \( 1 + 183.T + 2.97e4T^{2} \)
37 \( 1 - 418.T + 5.06e4T^{2} \)
41 \( 1 - 142.T + 6.89e4T^{2} \)
43 \( 1 - 71.0T + 7.95e4T^{2} \)
47 \( 1 - 323.T + 1.03e5T^{2} \)
53 \( 1 - 25.1T + 1.48e5T^{2} \)
59 \( 1 + 684.T + 2.05e5T^{2} \)
61 \( 1 - 308.T + 2.26e5T^{2} \)
67 \( 1 + 672.T + 3.00e5T^{2} \)
71 \( 1 + 326.T + 3.57e5T^{2} \)
73 \( 1 - 24.3T + 3.89e5T^{2} \)
79 \( 1 + 166.T + 4.93e5T^{2} \)
83 \( 1 + 201.T + 5.71e5T^{2} \)
89 \( 1 + 108.T + 7.04e5T^{2} \)
97 \( 1 - 1.15e3T + 9.12e5T^{2} \)
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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

−8.245736428002200750136047556038, −7.57547644245008626663418372191, −7.38208841751880378861147131901, −6.01279702449559026804490910220, −5.12714336593394908950690043799, −4.32047385002478398692844630940, −3.33745089719184438097433035452, −2.75545666297865441718368239673, −0.958062341644736310805027065430, 0, 0.958062341644736310805027065430, 2.75545666297865441718368239673, 3.33745089719184438097433035452, 4.32047385002478398692844630940, 5.12714336593394908950690043799, 6.01279702449559026804490910220, 7.38208841751880378861147131901, 7.57547644245008626663418372191, 8.245736428002200750136047556038

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