Properties

Label 2-48e2-1.1-c3-0-111
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  − 3.46·5-s + 24.2·7-s + 48·11-s + 41.5·13-s − 54·17-s + 4·19-s − 173.·23-s − 113·25-s − 162.·29-s − 58.8·31-s − 84·35-s − 325.·37-s − 294·41-s − 188·43-s + 505.·47-s + 245·49-s − 744.·53-s − 166.·55-s + 252·59-s − 90.0·61-s − 144·65-s − 628·67-s + 6.92·71-s + 1.00e3·73-s + 1.16e3·77-s + 1.34e3·79-s − 720·83-s + ⋯
L(s)  = 1  − 0.309·5-s + 1.30·7-s + 1.31·11-s + 0.886·13-s − 0.770·17-s + 0.0482·19-s − 1.57·23-s − 0.904·25-s − 1.04·29-s − 0.341·31-s − 0.405·35-s − 1.44·37-s − 1.11·41-s − 0.666·43-s + 1.56·47-s + 0.714·49-s − 1.93·53-s − 0.407·55-s + 0.556·59-s − 0.189·61-s − 0.274·65-s − 1.14·67-s + 0.0115·71-s + 1.61·73-s + 1.72·77-s + 1.90·79-s − 0.952·83-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 \( 1 + 3.46T + 125T^{2} \)
7 \( 1 - 24.2T + 343T^{2} \)
11 \( 1 - 48T + 1.33e3T^{2} \)
13 \( 1 - 41.5T + 2.19e3T^{2} \)
17 \( 1 + 54T + 4.91e3T^{2} \)
19 \( 1 - 4T + 6.85e3T^{2} \)
23 \( 1 + 173.T + 1.21e4T^{2} \)
29 \( 1 + 162.T + 2.43e4T^{2} \)
31 \( 1 + 58.8T + 2.97e4T^{2} \)
37 \( 1 + 325.T + 5.06e4T^{2} \)
41 \( 1 + 294T + 6.89e4T^{2} \)
43 \( 1 + 188T + 7.95e4T^{2} \)
47 \( 1 - 505.T + 1.03e5T^{2} \)
53 \( 1 + 744.T + 1.48e5T^{2} \)
59 \( 1 - 252T + 2.05e5T^{2} \)
61 \( 1 + 90.0T + 2.26e5T^{2} \)
67 \( 1 + 628T + 3.00e5T^{2} \)
71 \( 1 - 6.92T + 3.57e5T^{2} \)
73 \( 1 - 1.00e3T + 3.89e5T^{2} \)
79 \( 1 - 1.34e3T + 4.93e5T^{2} \)
83 \( 1 + 720T + 5.71e5T^{2} \)
89 \( 1 + 1.48e3T + 7.04e5T^{2} \)
97 \( 1 - 1.82e3T + 9.12e5T^{2} \)
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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

−8.299746824443762797205188658549, −7.62780461239997810080473617193, −6.70297033047217251026603184311, −5.95266475950881671857631705078, −5.03677594889006977794378109400, −4.05932388802803264680483267752, −3.65744518810836359808686270632, −1.96570055431332706010856733659, −1.47597195929337093226813735880, 0, 1.47597195929337093226813735880, 1.96570055431332706010856733659, 3.65744518810836359808686270632, 4.05932388802803264680483267752, 5.03677594889006977794378109400, 5.95266475950881671857631705078, 6.70297033047217251026603184311, 7.62780461239997810080473617193, 8.299746824443762797205188658549

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