Properties

Label 2-48e2-4.3-c2-0-23
Degree $2$
Conductor $2304$
Sign $1$
Analytic cond. $62.7794$
Root an. cond. $7.92334$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 24·13-s + 16·17-s + 11·25-s − 42·29-s + 24·37-s − 80·41-s + 49·49-s − 90·53-s + 120·61-s + 144·65-s + 110·73-s − 96·85-s − 160·89-s + 130·97-s + 198·101-s − 120·109-s + 224·113-s + ⋯
L(s)  = 1  − 6/5·5-s − 1.84·13-s + 0.941·17-s + 0.439·25-s − 1.44·29-s + 0.648·37-s − 1.95·41-s + 49-s − 1.69·53-s + 1.96·61-s + 2.21·65-s + 1.50·73-s − 1.12·85-s − 1.79·89-s + 1.34·97-s + 1.96·101-s − 1.10·109-s + 1.98·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2304} (1279, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8944955646\)
\(L(\frac12)\) \(\approx\) \(0.8944955646\)
\(L(2)\) 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 \( 1 + 6 T + p^{2} T^{2} \)
7 \( ( 1 - p T )( 1 + p T ) \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 + 24 T + p^{2} T^{2} \)
17 \( 1 - 16 T + p^{2} T^{2} \)
19 \( ( 1 - p T )( 1 + p T ) \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( 1 + 42 T + p^{2} T^{2} \)
31 \( ( 1 - p T )( 1 + p T ) \)
37 \( 1 - 24 T + p^{2} T^{2} \)
41 \( 1 + 80 T + p^{2} T^{2} \)
43 \( ( 1 - p T )( 1 + p T ) \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( 1 + 90 T + p^{2} T^{2} \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 - 120 T + p^{2} T^{2} \)
67 \( ( 1 - p T )( 1 + p T ) \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 - 110 T + p^{2} T^{2} \)
79 \( ( 1 - p T )( 1 + p T ) \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( 1 + 160 T + p^{2} T^{2} \)
97 \( 1 - 130 T + p^{2} T^{2} \)
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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

−8.731931215410081020618166246343, −7.81261977694033713446807476831, −7.50728947481209324457190485999, −6.71684368489088035558960355535, −5.49638032536542519672477746773, −4.83922007093774589206459198684, −3.91096068054447725702859093488, −3.14429910710236381825539193641, −2.01417992920065883921986664188, −0.47102575647765960743785862222, 0.47102575647765960743785862222, 2.01417992920065883921986664188, 3.14429910710236381825539193641, 3.91096068054447725702859093488, 4.83922007093774589206459198684, 5.49638032536542519672477746773, 6.71684368489088035558960355535, 7.50728947481209324457190485999, 7.81261977694033713446807476831, 8.731931215410081020618166246343

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