Properties

Label 2-47096-1.1-c1-0-10
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  − 3·3-s + 3·5-s + 7-s + 6·9-s − 3·11-s + 5·13-s − 9·15-s + 2·17-s − 4·19-s − 3·21-s + 4·23-s + 4·25-s − 9·27-s + 7·31-s + 9·33-s + 3·35-s − 2·37-s − 15·39-s − 8·41-s + 43-s + 18·45-s + 3·47-s + 49-s − 6·51-s + 9·53-s − 9·55-s + 12·57-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.34·5-s + 0.377·7-s + 2·9-s − 0.904·11-s + 1.38·13-s − 2.32·15-s + 0.485·17-s − 0.917·19-s − 0.654·21-s + 0.834·23-s + 4/5·25-s − 1.73·27-s + 1.25·31-s + 1.56·33-s + 0.507·35-s − 0.328·37-s − 2.40·39-s − 1.24·41-s + 0.152·43-s + 2.68·45-s + 0.437·47-s + 1/7·49-s − 0.840·51-s + 1.23·53-s − 1.21·55-s + 1.58·57-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 \( 1 + p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−15.10616857924049, −14.12958096491944, −13.64192659247953, −13.27255900579992, −12.80859460432238, −12.26795152819924, −11.67372753139254, −11.14691373574203, −10.69175067033211, −10.24964347887272, −10.03746801734875, −9.153705016656741, −8.562649000901140, −8.021440956961524, −7.081930634296512, −6.695591840152205, −6.074621038970506, −5.691636510468434, −5.338026118212281, −4.687138435320905, −4.138790500033522, −3.126309861263141, −2.320520303420172, −1.458852548347202, −1.076385564305581, 0, 1.076385564305581, 1.458852548347202, 2.320520303420172, 3.126309861263141, 4.138790500033522, 4.687138435320905, 5.338026118212281, 5.691636510468434, 6.074621038970506, 6.695591840152205, 7.081930634296512, 8.021440956961524, 8.562649000901140, 9.153705016656741, 10.03746801734875, 10.24964347887272, 10.69175067033211, 11.14691373574203, 11.67372753139254, 12.26795152819924, 12.80859460432238, 13.27255900579992, 13.64192659247953, 14.12958096491944, 15.10616857924049

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