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Properties

Label 2-139650-1.1-c1-0-114
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 + 5·11-s − 12-s − 13-s + 16-s + 6·17-s + 18-s + 19-s + 5·22-s − 23-s − 24-s − 26-s − 27-s + 4·29-s + 7·31-s + 32-s − 5·33-s + 6·34-s + 36-s + 2·37-s + 38-s + 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 + 1.50·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.229·19-s + 1.06·22-s − 0.208·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.742·29-s + 1.25·31-s + 0.176·32-s − 0.870·33-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.162·38-s + 0.160·39-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.279926307\)
\(L(\frac12)\) \(\approx\) \(5.279926307\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.60934425385440, −12.75253183894263, −12.33595748064428, −12.02244230536080, −11.74653583969527, −11.07093966402953, −10.73526968315107, −9.997790237477462, −9.639589813548863, −9.279329037897892, −8.398211945276767, −7.961906123830633, −7.430044982913708, −6.799063353158452, −6.423534330379226, −5.935978628236178, −5.494657725085059, −4.785287140261403, −4.398820957771624, −3.840533110807327, −3.246821191702333, −2.702440424867850, −1.842110450826157, −1.153340935157509, −0.7230826473158094, 0.7230826473158094, 1.153340935157509, 1.842110450826157, 2.702440424867850, 3.246821191702333, 3.840533110807327, 4.398820957771624, 4.785287140261403, 5.494657725085059, 5.935978628236178, 6.423534330379226, 6.799063353158452, 7.430044982913708, 7.961906123830633, 8.398211945276767, 9.279329037897892, 9.639589813548863, 9.997790237477462, 10.73526968315107, 11.07093966402953, 11.74653583969527, 12.02244230536080, 12.33595748064428, 12.75253183894263, 13.60934425385440

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