Properties

Label 2-1840-1.1-c3-0-22
Degree $2$
Conductor $1840$
Sign $1$
Analytic cond. $108.563$
Root an. cond. $10.4193$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.73·3-s − 5·5-s − 23.5·7-s + 32.7·9-s + 32.1·11-s + 40.0·13-s + 38.6·15-s + 126.·17-s − 0.232·19-s + 182.·21-s + 23·23-s + 25·25-s − 44.5·27-s − 137.·29-s − 112.·31-s − 248.·33-s + 117.·35-s + 45.7·37-s − 309.·39-s − 135.·41-s − 543.·43-s − 163.·45-s − 26.4·47-s + 212.·49-s − 975.·51-s + 43.6·53-s − 160.·55-s + ⋯
L(s)  = 1  − 1.48·3-s − 0.447·5-s − 1.27·7-s + 1.21·9-s + 0.880·11-s + 0.855·13-s + 0.665·15-s + 1.79·17-s − 0.00281·19-s + 1.89·21-s + 0.208·23-s + 0.200·25-s − 0.317·27-s − 0.878·29-s − 0.653·31-s − 1.31·33-s + 0.568·35-s + 0.203·37-s − 1.27·39-s − 0.515·41-s − 1.92·43-s − 0.542·45-s − 0.0820·47-s + 0.618·49-s − 2.67·51-s + 0.113·53-s − 0.393·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(108.563\)
Root analytic conductor: \(10.4193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8045128434\)
\(L(\frac12)\) \(\approx\) \(0.8045128434\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 5T \)
23 \( 1 - 23T \)
good3 \( 1 + 7.73T + 27T^{2} \)
7 \( 1 + 23.5T + 343T^{2} \)
11 \( 1 - 32.1T + 1.33e3T^{2} \)
13 \( 1 - 40.0T + 2.19e3T^{2} \)
17 \( 1 - 126.T + 4.91e3T^{2} \)
19 \( 1 + 0.232T + 6.85e3T^{2} \)
29 \( 1 + 137.T + 2.43e4T^{2} \)
31 \( 1 + 112.T + 2.97e4T^{2} \)
37 \( 1 - 45.7T + 5.06e4T^{2} \)
41 \( 1 + 135.T + 6.89e4T^{2} \)
43 \( 1 + 543.T + 7.95e4T^{2} \)
47 \( 1 + 26.4T + 1.03e5T^{2} \)
53 \( 1 - 43.6T + 1.48e5T^{2} \)
59 \( 1 + 202.T + 2.05e5T^{2} \)
61 \( 1 - 150.T + 2.26e5T^{2} \)
67 \( 1 - 420.T + 3.00e5T^{2} \)
71 \( 1 + 667.T + 3.57e5T^{2} \)
73 \( 1 - 602.T + 3.89e5T^{2} \)
79 \( 1 - 1.37e3T + 4.93e5T^{2} \)
83 \( 1 - 485.T + 5.71e5T^{2} \)
89 \( 1 + 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 1.48e3T + 9.12e5T^{2} \)
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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

−9.060028897535342528541398842910, −7.986083057677233223276951786722, −6.99174934272841970700791005633, −6.43281136248278194634687419635, −5.77886825034172779044131300122, −5.04413336384957841475612249174, −3.77826805176956558262818533438, −3.32705388150961290378074526129, −1.41878223218199821610406992081, −0.49462493273344896557532024330, 0.49462493273344896557532024330, 1.41878223218199821610406992081, 3.32705388150961290378074526129, 3.77826805176956558262818533438, 5.04413336384957841475612249174, 5.77886825034172779044131300122, 6.43281136248278194634687419635, 6.99174934272841970700791005633, 7.986083057677233223276951786722, 9.060028897535342528541398842910

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