Properties

Label 2-139650-1.1-c1-0-132
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 + 2·11-s + 12-s + 13-s + 16-s + 18-s − 19-s + 2·22-s + 8·23-s + 24-s + 26-s + 27-s + 10·29-s + 8·31-s + 32-s + 2·33-s + 36-s − 4·37-s − 38-s + 39-s + 5·41-s − 8·43-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 + 0.603·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.235·18-s − 0.229·19-s + 0.426·22-s + 1.66·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.176·32-s + 0.348·33-s + 1/6·36-s − 0.657·37-s − 0.162·38-s + 0.160·39-s + 0.780·41-s − 1.21·43-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

\(L(1)\) \(\approx\) \(7.784957234\)
\(L(\frac12)\) \(\approx\) \(7.784957234\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.60862563987772, −12.96708748048067, −12.47215674345962, −12.13365029999538, −11.60552308596633, −10.99912033155541, −10.66880987638394, −10.01666563974852, −9.608282262131815, −8.957513810208728, −8.472162876669553, −8.189616873778196, −7.393888990414382, −6.902847025053921, −6.536599503157864, −6.036513004339087, −5.317391955565131, −4.620126375196535, −4.503011951878676, −3.678221374619406, −3.122422157430666, −2.764431572496421, −2.045694291499211, −1.252847651379002, −0.7716031325923583, 0.7716031325923583, 1.252847651379002, 2.045694291499211, 2.764431572496421, 3.122422157430666, 3.678221374619406, 4.503011951878676, 4.620126375196535, 5.317391955565131, 6.036513004339087, 6.536599503157864, 6.902847025053921, 7.393888990414382, 8.189616873778196, 8.472162876669553, 8.957513810208728, 9.608282262131815, 10.01666563974852, 10.66880987638394, 10.99912033155541, 11.60552308596633, 12.13365029999538, 12.47215674345962, 12.96708748048067, 13.60862563987772

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