Properties

Label 2-139650-1.1-c1-0-123
Degree $2$
Conductor $139650$
Sign $1$
Analytic cond. $1115.11$
Root an. cond. $33.3932$
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-s + 3-s + 4-s + 6-s + 8-s + 9-s + 11-s + 12-s + 4·13-s + 16-s + 4·17-s + 18-s + 19-s + 22-s − 5·23-s + 24-s + 4·26-s + 27-s + 3·29-s + 5·31-s + 32-s + 33-s + 4·34-s + 36-s + 6·37-s + 38-s + 4·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.229·19-s + 0.213·22-s − 1.04·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.557·29-s + 0.898·31-s + 0.176·32-s + 0.174·33-s + 0.685·34-s + 1/6·36-s + 0.986·37-s + 0.162·38-s + 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(139650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(1115.11\)
Root analytic conductor: \(33.3932\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{139650} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.624103955\)
\(L(\frac12)\) \(\approx\) \(7.624103955\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
19 \( 1 - T \)
good11 \( 1 - T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.53768066823872, −12.97416196643910, −12.57335676068633, −11.99343552186812, −11.61270127688428, −11.15894489316333, −10.48858391786639, −10.06738439982546, −9.624953582921813, −9.040235177996124, −8.414034089587345, −7.971745048413106, −7.692530861665541, −6.898524477160396, −6.314643477625740, −6.108812143969349, −5.377730601081806, −4.816972105896643, −4.204871216032338, −3.696463474528580, −3.297178536501548, −2.661146917799192, −2.015483268171122, −1.309137371478539, −0.7435090151928601, 0.7435090151928601, 1.309137371478539, 2.015483268171122, 2.661146917799192, 3.297178536501548, 3.696463474528580, 4.204871216032338, 4.816972105896643, 5.377730601081806, 6.108812143969349, 6.314643477625740, 6.898524477160396, 7.692530861665541, 7.971745048413106, 8.414034089587345, 9.040235177996124, 9.624953582921813, 10.06738439982546, 10.48858391786639, 11.15894489316333, 11.61270127688428, 11.99343552186812, 12.57335676068633, 12.97416196643910, 13.53768066823872

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