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Properties

Label 2-273e2-1.1-c1-0-19
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 + 19-s − 5·25-s + 7·31-s + 11·37-s + 8·43-s + 61-s − 8·64-s + 11·67-s + 10·73-s − 2·76-s − 13·79-s + 19·97-s + 10·100-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 4-s + 16-s + 0.229·19-s − 25-s + 1.25·31-s + 1.80·37-s + 1.21·43-s + 0.128·61-s − 64-s + 1.34·67-s + 1.17·73-s − 0.229·76-s − 1.46·79-s + 1.92·97-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

\(L(1)\) \(\approx\) \(1.881803409\)
\(L(\frac12)\) \(\approx\) \(1.881803409\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−14.05490466125991, −13.56548960564602, −13.14497279484900, −12.65775895757643, −12.15384128672060, −11.58676088831169, −11.12352060392393, −10.44158756490750, −9.921988292170818, −9.523311892504140, −9.123226581680478, −8.410567587002503, −7.982416727631741, −7.617799451789624, −6.822282350050689, −6.184052378044581, −5.685128775481596, −5.178392040272860, −4.340387306108467, −4.237443850985628, −3.417285041073450, −2.770994969358157, −2.055015148227441, −1.073624207259623, −0.5426794770129261, 0.5426794770129261, 1.073624207259623, 2.055015148227441, 2.770994969358157, 3.417285041073450, 4.237443850985628, 4.340387306108467, 5.178392040272860, 5.685128775481596, 6.184052378044581, 6.822282350050689, 7.617799451789624, 7.982416727631741, 8.410567587002503, 9.123226581680478, 9.523311892504140, 9.921988292170818, 10.44158756490750, 11.12352060392393, 11.58676088831169, 12.15384128672060, 12.65775895757643, 13.14497279484900, 13.56548960564602, 14.05490466125991

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