Properties

Label 2-48e2-1.1-c1-0-24
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·7-s + 4·13-s + 2·17-s + 4·19-s − 4·23-s − 25-s − 6·29-s − 2·31-s + 4·35-s + 8·37-s + 2·41-s + 4·43-s − 12·47-s − 3·49-s − 6·53-s + 4·59-s − 8·65-s − 12·67-s − 12·71-s + 6·73-s − 10·79-s − 16·83-s − 4·85-s − 10·89-s − 8·91-s − 8·95-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.755·7-s + 1.10·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s − 0.359·31-s + 0.676·35-s + 1.31·37-s + 0.312·41-s + 0.609·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.520·59-s − 0.992·65-s − 1.46·67-s − 1.42·71-s + 0.702·73-s − 1.12·79-s − 1.75·83-s − 0.433·85-s − 1.05·89-s − 0.838·91-s − 0.820·95-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.516459749387738615660582919737, −7.82480057888576208813073884026, −7.22882752368325973405978234590, −6.18041387184693344257457765692, −5.65428990986115338832346663429, −4.38034697737372143869607214761, −3.66069496333291735371153326487, −2.98778579681725708048014017079, −1.44397590722771000597851729066, 0, 1.44397590722771000597851729066, 2.98778579681725708048014017079, 3.66069496333291735371153326487, 4.38034697737372143869607214761, 5.65428990986115338832346663429, 6.18041387184693344257457765692, 7.22882752368325973405978234590, 7.82480057888576208813073884026, 8.516459749387738615660582919737

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