Properties

Label 2-48e2-1.1-c3-0-89
Degree $2$
Conductor $2304$
Sign $-1$
Analytic cond. $135.940$
Root an. cond. $11.6593$
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  + 6·5-s − 21.1·7-s + 42.3·11-s + 20·13-s − 8·17-s + 84.6·19-s − 169.·23-s − 89·25-s − 46·29-s − 21.1·31-s − 126.·35-s − 164·37-s − 312·41-s + 423.·43-s − 169.·47-s + 105.·49-s + 266·53-s + 253.·55-s − 253.·59-s − 132·61-s + 120·65-s + 507.·67-s + 677.·71-s + 246·73-s − 896.·77-s + 232.·79-s + 973.·83-s + ⋯
L(s)  = 1  + 0.536·5-s − 1.14·7-s + 1.16·11-s + 0.426·13-s − 0.114·17-s + 1.02·19-s − 1.53·23-s − 0.711·25-s − 0.294·29-s − 0.122·31-s − 0.613·35-s − 0.728·37-s − 1.18·41-s + 1.50·43-s − 0.525·47-s + 0.306·49-s + 0.689·53-s + 0.622·55-s − 0.560·59-s − 0.277·61-s + 0.228·65-s + 0.926·67-s + 1.13·71-s + 0.394·73-s − 1.32·77-s + 0.331·79-s + 1.28·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(135.940\)
Root analytic conductor: \(11.6593\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2304,\ (\ :3/2),\ -1)\)

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 \( 1 \)
good5 \( 1 - 6T + 125T^{2} \)
7 \( 1 + 21.1T + 343T^{2} \)
11 \( 1 - 42.3T + 1.33e3T^{2} \)
13 \( 1 - 20T + 2.19e3T^{2} \)
17 \( 1 + 8T + 4.91e3T^{2} \)
19 \( 1 - 84.6T + 6.85e3T^{2} \)
23 \( 1 + 169.T + 1.21e4T^{2} \)
29 \( 1 + 46T + 2.43e4T^{2} \)
31 \( 1 + 21.1T + 2.97e4T^{2} \)
37 \( 1 + 164T + 5.06e4T^{2} \)
41 \( 1 + 312T + 6.89e4T^{2} \)
43 \( 1 - 423.T + 7.95e4T^{2} \)
47 \( 1 + 169.T + 1.03e5T^{2} \)
53 \( 1 - 266T + 1.48e5T^{2} \)
59 \( 1 + 253.T + 2.05e5T^{2} \)
61 \( 1 + 132T + 2.26e5T^{2} \)
67 \( 1 - 507.T + 3.00e5T^{2} \)
71 \( 1 - 677.T + 3.57e5T^{2} \)
73 \( 1 - 246T + 3.89e5T^{2} \)
79 \( 1 - 232.T + 4.93e5T^{2} \)
83 \( 1 - 973.T + 5.71e5T^{2} \)
89 \( 1 + 1.39e3T + 7.04e5T^{2} \)
97 \( 1 + 302T + 9.12e5T^{2} \)
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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

−8.352853269697175036416224998437, −7.40548272531159595479192813363, −6.52183601045389796271688640757, −6.09546471330166877581861176464, −5.26164519607894616552902495729, −3.93630733941639700700117854415, −3.49327041316085531818569445788, −2.27203965202186773816555075488, −1.27183521030992646316492767112, 0, 1.27183521030992646316492767112, 2.27203965202186773816555075488, 3.49327041316085531818569445788, 3.93630733941639700700117854415, 5.26164519607894616552902495729, 6.09546471330166877581861176464, 6.52183601045389796271688640757, 7.40548272531159595479192813363, 8.352853269697175036416224998437

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