Properties

Label 2-48e2-1.1-c3-0-53
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  − 22·5-s − 92·13-s + 104·17-s + 359·25-s − 130·29-s + 396·37-s + 472·41-s − 343·49-s + 518·53-s − 468·61-s + 2.02e3·65-s − 1.09e3·73-s − 2.28e3·85-s + 176·89-s + 594·97-s + 598·101-s + 1.46e3·109-s + 1.32e3·113-s + ⋯
L(s)  = 1  − 1.96·5-s − 1.96·13-s + 1.48·17-s + 2.87·25-s − 0.832·29-s + 1.75·37-s + 1.79·41-s − 49-s + 1.34·53-s − 0.982·61-s + 3.86·65-s − 1.76·73-s − 2.91·85-s + 0.209·89-s + 0.621·97-s + 0.589·101-s + 1.28·109-s + 1.10·113-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 \( 1 + 22 T + p^{3} T^{2} \)
7 \( 1 + p^{3} T^{2} \)
11 \( 1 + p^{3} T^{2} \)
13 \( 1 + 92 T + p^{3} T^{2} \)
17 \( 1 - 104 T + p^{3} T^{2} \)
19 \( 1 + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + 130 T + p^{3} T^{2} \)
31 \( 1 + p^{3} T^{2} \)
37 \( 1 - 396 T + p^{3} T^{2} \)
41 \( 1 - 472 T + p^{3} T^{2} \)
43 \( 1 + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 - 518 T + p^{3} T^{2} \)
59 \( 1 + p^{3} T^{2} \)
61 \( 1 + 468 T + p^{3} T^{2} \)
67 \( 1 + p^{3} T^{2} \)
71 \( 1 + p^{3} T^{2} \)
73 \( 1 + 1098 T + p^{3} T^{2} \)
79 \( 1 + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 - 176 T + p^{3} T^{2} \)
97 \( 1 - 594 T + p^{3} T^{2} \)
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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

−7.894867912411931703206274951198, −7.62876930548703307555672137437, −7.12925052998838309102803294523, −5.85440156707547531551726609467, −4.84109160943709746064046037542, −4.26864956673442495466280777739, −3.35559673055214973587538437005, −2.56865777988317765557524570234, −0.906534888144040192744528763917, 0, 0.906534888144040192744528763917, 2.56865777988317765557524570234, 3.35559673055214973587538437005, 4.26864956673442495466280777739, 4.84109160943709746064046037542, 5.85440156707547531551726609467, 7.12925052998838309102803294523, 7.62876930548703307555672137437, 7.894867912411931703206274951198

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