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Properties

Label 2-273e2-1.1-c1-0-3
Degree $2$
Conductor $74529$
Sign $1$
Analytic cond. $595.117$
Root an. cond. $24.3950$
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  − 2·4-s + 4·16-s − 8·19-s − 5·25-s + 7·31-s − 10·37-s − 13·43-s + 13·61-s − 8·64-s + 11·67-s − 17·73-s + 16·76-s − 13·79-s − 5·97-s + 10·100-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 4-s + 16-s − 1.83·19-s − 25-s + 1.25·31-s − 1.64·37-s − 1.98·43-s + 1.66·61-s − 64-s + 1.34·67-s − 1.98·73-s + 1.83·76-s − 1.46·79-s − 0.507·97-s + 100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74529\)    =    \(3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(595.117\)
Root analytic conductor: \(24.3950\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 74529,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4183699216\)
\(L(\frac12)\) \(\approx\) \(0.4183699216\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−14.03209772885799, −13.47575614741749, −13.17727574556050, −12.70069937557007, −12.08204883844804, −11.70452176756340, −11.04022141101352, −10.33140973135303, −10.07894575694242, −9.638731795479580, −8.798778338964359, −8.487977205795018, −8.230648435330086, −7.449757595552249, −6.758383996759658, −6.327940640177495, −5.645018396181727, −5.100633526562951, −4.556894202448686, −3.987259533936207, −3.557439146506554, −2.729903379202371, −1.971475664677240, −1.291003495231820, −0.2215034072698074, 0.2215034072698074, 1.291003495231820, 1.971475664677240, 2.729903379202371, 3.557439146506554, 3.987259533936207, 4.556894202448686, 5.100633526562951, 5.645018396181727, 6.327940640177495, 6.758383996759658, 7.449757595552249, 8.230648435330086, 8.487977205795018, 8.798778338964359, 9.638731795479580, 10.07894575694242, 10.33140973135303, 11.04022141101352, 11.70452176756340, 12.08204883844804, 12.70069937557007, 13.17727574556050, 13.47575614741749, 14.03209772885799

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