Properties

Label 2-48e2-1.1-c1-0-11
Degree $2$
Conductor $2304$
Sign $1$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·11-s + 6·17-s − 2·19-s − 5·25-s − 6·41-s + 10·43-s − 7·49-s + 6·59-s + 14·67-s − 2·73-s + 18·83-s + 18·89-s + 10·97-s + 6·107-s − 18·113-s + ⋯
L(s)  = 1  + 1.80·11-s + 1.45·17-s − 0.458·19-s − 25-s − 0.937·41-s + 1.52·43-s − 49-s + 0.781·59-s + 1.71·67-s − 0.234·73-s + 1.97·83-s + 1.90·89-s + 1.01·97-s + 0.580·107-s − 1.69·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2304} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.056695321\)
\(L(\frac12)\) \(\approx\) \(2.056695321\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.111811254215445438085933089837, −8.227029907063207442212793556286, −7.49800309626174308942277426392, −6.58492595927970716736421063363, −6.00725685310885624467606512287, −5.04834818214897768625909783776, −3.97459158177538150872708928083, −3.44733369104578574594522067100, −2.04286410288724293558493803436, −0.986409188428603835741707173143, 0.986409188428603835741707173143, 2.04286410288724293558493803436, 3.44733369104578574594522067100, 3.97459158177538150872708928083, 5.04834818214897768625909783776, 6.00725685310885624467606512287, 6.58492595927970716736421063363, 7.49800309626174308942277426392, 8.227029907063207442212793556286, 9.111811254215445438085933089837

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