Properties

Label 2-48e2-1.1-c3-0-70
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  − 10.3·5-s − 3.46·7-s − 55.4·13-s + 90·17-s − 116·19-s + 103.·23-s − 17·25-s + 259.·29-s + 301.·31-s + 36·35-s + 34.6·37-s − 54·41-s − 20·43-s + 394.·47-s − 331·49-s − 488.·53-s + 324·59-s + 575.·61-s + 576·65-s + 116·67-s − 1.10e3·71-s − 1.10e3·73-s + 148.·79-s − 1.15e3·83-s − 935.·85-s + 918·89-s + 191.·91-s + ⋯
L(s)  = 1  − 0.929·5-s − 0.187·7-s − 1.18·13-s + 1.28·17-s − 1.40·19-s + 0.942·23-s − 0.136·25-s + 1.66·29-s + 1.74·31-s + 0.173·35-s + 0.153·37-s − 0.205·41-s − 0.0709·43-s + 1.22·47-s − 0.965·49-s − 1.26·53-s + 0.714·59-s + 1.20·61-s + 1.09·65-s + 0.211·67-s − 1.84·71-s − 1.77·73-s + 0.212·79-s − 1.52·83-s − 1.19·85-s + 1.09·89-s + 0.221·91-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 \( 1 + 10.3T + 125T^{2} \)
7 \( 1 + 3.46T + 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 + 55.4T + 2.19e3T^{2} \)
17 \( 1 - 90T + 4.91e3T^{2} \)
19 \( 1 + 116T + 6.85e3T^{2} \)
23 \( 1 - 103.T + 1.21e4T^{2} \)
29 \( 1 - 259.T + 2.43e4T^{2} \)
31 \( 1 - 301.T + 2.97e4T^{2} \)
37 \( 1 - 34.6T + 5.06e4T^{2} \)
41 \( 1 + 54T + 6.89e4T^{2} \)
43 \( 1 + 20T + 7.95e4T^{2} \)
47 \( 1 - 394.T + 1.03e5T^{2} \)
53 \( 1 + 488.T + 1.48e5T^{2} \)
59 \( 1 - 324T + 2.05e5T^{2} \)
61 \( 1 - 575.T + 2.26e5T^{2} \)
67 \( 1 - 116T + 3.00e5T^{2} \)
71 \( 1 + 1.10e3T + 3.57e5T^{2} \)
73 \( 1 + 1.10e3T + 3.89e5T^{2} \)
79 \( 1 - 148.T + 4.93e5T^{2} \)
83 \( 1 + 1.15e3T + 5.71e5T^{2} \)
89 \( 1 - 918T + 7.04e5T^{2} \)
97 \( 1 - 190T + 9.12e5T^{2} \)
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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

−8.219196758602056358910343123426, −7.55954017117474441047608057536, −6.81674433646148550979848336216, −5.99792553621883396640395132025, −4.85234993498273774135704224725, −4.35372904206542681601505199800, −3.24860052789418178953572526777, −2.51368577855479832840806227120, −1.04645410159581640400642783804, 0, 1.04645410159581640400642783804, 2.51368577855479832840806227120, 3.24860052789418178953572526777, 4.35372904206542681601505199800, 4.85234993498273774135704224725, 5.99792553621883396640395132025, 6.81674433646148550979848336216, 7.55954017117474441047608057536, 8.219196758602056358910343123426

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