Properties

Label 2-47096-1.1-c1-0-4
Degree $2$
Conductor $47096$
Sign $-1$
Analytic cond. $376.063$
Root an. cond. $19.3923$
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  − 3-s + 5-s + 7-s − 2·9-s − 4·11-s − 5·13-s − 15-s − 4·17-s + 19-s − 21-s − 23-s − 4·25-s + 5·27-s + 4·31-s + 4·33-s + 35-s + 10·37-s + 5·39-s + 6·41-s − 8·43-s − 2·45-s − 4·47-s + 49-s + 4·51-s + 8·53-s − 4·55-s − 57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.20·11-s − 1.38·13-s − 0.258·15-s − 0.970·17-s + 0.229·19-s − 0.218·21-s − 0.208·23-s − 4/5·25-s + 0.962·27-s + 0.718·31-s + 0.696·33-s + 0.169·35-s + 1.64·37-s + 0.800·39-s + 0.937·41-s − 1.21·43-s − 0.298·45-s − 0.583·47-s + 1/7·49-s + 0.560·51-s + 1.09·53-s − 0.539·55-s − 0.132·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47096\)    =    \(2^{3} \cdot 7 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(376.063\)
Root analytic conductor: \(19.3923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{47096} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 47096,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
29 \( 1 \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−14.89640559702898, −14.32721371708640, −13.80313336289198, −13.25491867109501, −12.87282338069549, −12.19719554335353, −11.63990264510841, −11.36186488269093, −10.70704310410659, −10.16738353169468, −9.772938010352427, −9.170680433910475, −8.454609687462373, −7.932042178317187, −7.521444569746741, −6.736858381278631, −6.225886790672433, −5.574355406373187, −5.154300417921431, −4.682553121270946, −4.011408777197177, −2.886009021965268, −2.504277193849026, −1.955711746251645, −0.7435015331029543, 0, 0.7435015331029543, 1.955711746251645, 2.504277193849026, 2.886009021965268, 4.011408777197177, 4.682553121270946, 5.154300417921431, 5.574355406373187, 6.225886790672433, 6.736858381278631, 7.521444569746741, 7.932042178317187, 8.454609687462373, 9.170680433910475, 9.772938010352427, 10.16738353169468, 10.70704310410659, 11.36186488269093, 11.63990264510841, 12.19719554335353, 12.87282338069549, 13.25491867109501, 13.80313336289198, 14.32721371708640, 14.89640559702898

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