Properties

Label 2-1872-1.1-c3-0-45
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  − 14.0·5-s − 24.2·7-s + 3.10·11-s − 13·13-s − 43.9·17-s + 85.8·19-s + 203.·23-s + 73.4·25-s + 31.0·29-s + 135.·31-s + 341.·35-s + 290.·37-s + 148.·41-s − 281.·43-s − 225.·47-s + 245.·49-s − 172.·53-s − 43.7·55-s − 41.2·59-s + 499.·61-s + 183.·65-s − 503.·67-s + 946.·71-s − 1.11e3·73-s − 75.3·77-s − 674.·79-s + 59.4·83-s + ⋯
L(s)  = 1  − 1.25·5-s − 1.31·7-s + 0.0851·11-s − 0.277·13-s − 0.626·17-s + 1.03·19-s + 1.84·23-s + 0.587·25-s + 0.198·29-s + 0.786·31-s + 1.65·35-s + 1.28·37-s + 0.566·41-s − 0.996·43-s − 0.701·47-s + 0.716·49-s − 0.447·53-s − 0.107·55-s − 0.0909·59-s + 1.04·61-s + 0.349·65-s − 0.918·67-s + 1.58·71-s − 1.78·73-s − 0.111·77-s − 0.961·79-s + 0.0786·83-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 \( 1 + 14.0T + 125T^{2} \)
7 \( 1 + 24.2T + 343T^{2} \)
11 \( 1 - 3.10T + 1.33e3T^{2} \)
17 \( 1 + 43.9T + 4.91e3T^{2} \)
19 \( 1 - 85.8T + 6.85e3T^{2} \)
23 \( 1 - 203.T + 1.21e4T^{2} \)
29 \( 1 - 31.0T + 2.43e4T^{2} \)
31 \( 1 - 135.T + 2.97e4T^{2} \)
37 \( 1 - 290.T + 5.06e4T^{2} \)
41 \( 1 - 148.T + 6.89e4T^{2} \)
43 \( 1 + 281.T + 7.95e4T^{2} \)
47 \( 1 + 225.T + 1.03e5T^{2} \)
53 \( 1 + 172.T + 1.48e5T^{2} \)
59 \( 1 + 41.2T + 2.05e5T^{2} \)
61 \( 1 - 499.T + 2.26e5T^{2} \)
67 \( 1 + 503.T + 3.00e5T^{2} \)
71 \( 1 - 946.T + 3.57e5T^{2} \)
73 \( 1 + 1.11e3T + 3.89e5T^{2} \)
79 \( 1 + 674.T + 4.93e5T^{2} \)
83 \( 1 - 59.4T + 5.71e5T^{2} \)
89 \( 1 - 1.21e3T + 7.04e5T^{2} \)
97 \( 1 + 879.T + 9.12e5T^{2} \)
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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

−8.468909178207164798214389748841, −7.60624888209001107955132776229, −6.95318381335038903918581270020, −6.29145154370225907360820580309, −5.10490771161957135264713629681, −4.26924507493639001365615677424, −3.32708996039277804782914562409, −2.77219022289939646737603286087, −0.971236304285921129016295712155, 0, 0.971236304285921129016295712155, 2.77219022289939646737603286087, 3.32708996039277804782914562409, 4.26924507493639001365615677424, 5.10490771161957135264713629681, 6.29145154370225907360820580309, 6.95318381335038903918581270020, 7.60624888209001107955132776229, 8.468909178207164798214389748841

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