Properties

Label 2-2240-1.1-c3-0-143
Degree $2$
Conductor $2240$
Sign $-1$
Analytic cond. $132.164$
Root an. cond. $11.4962$
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  + 6.65·3-s + 5·5-s + 7·7-s + 17.3·9-s + 38.2·11-s − 19.3·13-s + 33.2·15-s − 87.2·17-s − 44.2·19-s + 46.5·21-s − 218.·23-s + 25·25-s − 64.4·27-s + 46.9·29-s − 194.·31-s + 254.·33-s + 35·35-s − 366.·37-s − 128.·39-s − 339.·41-s − 226.·43-s + 86.5·45-s − 11.6·47-s + 49·49-s − 580.·51-s + 209.·53-s + 191.·55-s + ⋯
L(s)  = 1  + 1.28·3-s + 0.447·5-s + 0.377·7-s + 0.641·9-s + 1.04·11-s − 0.412·13-s + 0.572·15-s − 1.24·17-s − 0.534·19-s + 0.484·21-s − 1.97·23-s + 0.200·25-s − 0.459·27-s + 0.300·29-s − 1.12·31-s + 1.34·33-s + 0.169·35-s − 1.63·37-s − 0.528·39-s − 1.29·41-s − 0.802·43-s + 0.286·45-s − 0.0362·47-s + 0.142·49-s − 1.59·51-s + 0.541·53-s + 0.468·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(132.164\)
Root analytic conductor: \(11.4962\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2240,\ (\ :3/2),\ -1)\)

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 \( 1 \)
5 \( 1 - 5T \)
7 \( 1 - 7T \)
good3 \( 1 - 6.65T + 27T^{2} \)
11 \( 1 - 38.2T + 1.33e3T^{2} \)
13 \( 1 + 19.3T + 2.19e3T^{2} \)
17 \( 1 + 87.2T + 4.91e3T^{2} \)
19 \( 1 + 44.2T + 6.85e3T^{2} \)
23 \( 1 + 218.T + 1.21e4T^{2} \)
29 \( 1 - 46.9T + 2.43e4T^{2} \)
31 \( 1 + 194.T + 2.97e4T^{2} \)
37 \( 1 + 366.T + 5.06e4T^{2} \)
41 \( 1 + 339.T + 6.89e4T^{2} \)
43 \( 1 + 226.T + 7.95e4T^{2} \)
47 \( 1 + 11.6T + 1.03e5T^{2} \)
53 \( 1 - 209.T + 1.48e5T^{2} \)
59 \( 1 + 616T + 2.05e5T^{2} \)
61 \( 1 + 320.T + 2.26e5T^{2} \)
67 \( 1 - 14.5T + 3.00e5T^{2} \)
71 \( 1 - 952T + 3.57e5T^{2} \)
73 \( 1 - 824.T + 3.89e5T^{2} \)
79 \( 1 + 156.T + 4.93e5T^{2} \)
83 \( 1 + 1.03e3T + 5.71e5T^{2} \)
89 \( 1 + 170.T + 7.04e5T^{2} \)
97 \( 1 - 1.05e3T + 9.12e5T^{2} \)
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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

−8.595870635896102958878377643746, −7.65988811618979051790030164718, −6.81341008253911706732969387774, −6.09137017874714797568270183216, −4.95170556172904331458375373407, −4.03947960390498919703307412064, −3.35430056655273078236514631588, −2.03437076767309082468813269112, −1.85785643775097848630525826702, 0, 1.85785643775097848630525826702, 2.03437076767309082468813269112, 3.35430056655273078236514631588, 4.03947960390498919703307412064, 4.95170556172904331458375373407, 6.09137017874714797568270183216, 6.81341008253911706732969387774, 7.65988811618979051790030164718, 8.595870635896102958878377643746

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