Properties

Label 2-29400-1.1-c1-0-3
Degree $2$
Conductor $29400$
Sign $1$
Analytic cond. $234.760$
Root an. cond. $15.3218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 11-s − 5·13-s − 4·17-s − 5·19-s + 3·23-s − 27-s + 8·29-s − 6·31-s + 33-s − 9·37-s + 5·39-s + 3·41-s + 4·43-s + 47-s + 4·51-s + 9·53-s + 5·57-s + 4·59-s + 4·61-s − 4·67-s − 3·69-s + 2·71-s − 14·73-s − 4·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.38·13-s − 0.970·17-s − 1.14·19-s + 0.625·23-s − 0.192·27-s + 1.48·29-s − 1.07·31-s + 0.174·33-s − 1.47·37-s + 0.800·39-s + 0.468·41-s + 0.609·43-s + 0.145·47-s + 0.560·51-s + 1.23·53-s + 0.662·57-s + 0.520·59-s + 0.512·61-s − 0.488·67-s − 0.361·69-s + 0.237·71-s − 1.63·73-s − 0.450·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29400\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(234.760\)
Root analytic conductor: \(15.3218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6106322491\)
\(L(\frac12)\) \(\approx\) \(0.6106322491\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−15.11420571661955, −14.72411150029692, −14.19090297948654, −13.42245366240953, −13.00939080275638, −12.41726574316656, −12.06508013060354, −11.44339312440720, −10.68647956145802, −10.52805806999190, −9.875730152762739, −9.144248826505335, −8.696070379880743, −8.057827295388304, −7.154282390482639, −6.999522336388667, −6.319328386308044, −5.507844080127933, −5.078102259185937, −4.407510013350851, −3.926961189319924, −2.764050346819582, −2.372044448272856, −1.449626567147284, −0.3074860697255823, 0.3074860697255823, 1.449626567147284, 2.372044448272856, 2.764050346819582, 3.926961189319924, 4.407510013350851, 5.078102259185937, 5.507844080127933, 6.319328386308044, 6.999522336388667, 7.154282390482639, 8.057827295388304, 8.696070379880743, 9.144248826505335, 9.875730152762739, 10.52805806999190, 10.68647956145802, 11.44339312440720, 12.06508013060354, 12.41726574316656, 13.00939080275638, 13.42245366240953, 14.19090297948654, 14.72411150029692, 15.11420571661955

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