Properties

Label 2-48e2-1.1-c3-0-86
Degree $2$
Conductor $2304$
Sign $-1$
Analytic cond. $135.940$
Root an. cond. $11.6593$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 18·11-s − 90·17-s + 106·19-s − 125·25-s + 522·41-s − 290·43-s − 343·49-s − 846·59-s − 70·67-s + 430·73-s + 1.35e3·83-s + 1.02e3·89-s − 1.91e3·97-s − 1.71e3·107-s + 270·113-s + ⋯
L(s)  = 1  + 0.493·11-s − 1.28·17-s + 1.27·19-s − 25-s + 1.98·41-s − 1.02·43-s − 49-s − 1.86·59-s − 0.127·67-s + 0.689·73-s + 1.78·83-s + 1.22·89-s − 1.99·97-s − 1.54·107-s + 0.224·113-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: yes
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 \( 1 + p^{3} T^{2} \)
7 \( 1 + p^{3} T^{2} \)
11 \( 1 - 18 T + p^{3} T^{2} \)
13 \( 1 + p^{3} T^{2} \)
17 \( 1 + 90 T + p^{3} T^{2} \)
19 \( 1 - 106 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + p^{3} T^{2} \)
31 \( 1 + p^{3} T^{2} \)
37 \( 1 + p^{3} T^{2} \)
41 \( 1 - 522 T + p^{3} T^{2} \)
43 \( 1 + 290 T + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 + p^{3} T^{2} \)
59 \( 1 + 846 T + p^{3} T^{2} \)
61 \( 1 + p^{3} T^{2} \)
67 \( 1 + 70 T + p^{3} T^{2} \)
71 \( 1 + p^{3} T^{2} \)
73 \( 1 - 430 T + p^{3} T^{2} \)
79 \( 1 + p^{3} T^{2} \)
83 \( 1 - 1350 T + p^{3} T^{2} \)
89 \( 1 - 1026 T + p^{3} T^{2} \)
97 \( 1 + 1910 T + p^{3} T^{2} \)
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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

−8.191132314343836152619901867272, −7.55594562761698016752431275780, −6.67123889792082147643897757983, −6.00756528946653799041147593200, −5.04516576862537076171631236981, −4.22819973099900732235591520751, −3.34400914920932583900853941132, −2.29293431684543282409752177693, −1.25272545831596136048797869530, 0, 1.25272545831596136048797869530, 2.29293431684543282409752177693, 3.34400914920932583900853941132, 4.22819973099900732235591520751, 5.04516576862537076171631236981, 6.00756528946653799041147593200, 6.67123889792082147643897757983, 7.55594562761698016752431275780, 8.191132314343836152619901867272

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