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Properties

Label 2-139650-1.1-c1-0-159
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 4·11-s − 12-s + 2·13-s + 16-s + 2·17-s − 18-s + 19-s − 4·22-s − 4·23-s + 24-s − 2·26-s − 27-s + 6·29-s − 4·31-s − 32-s − 4·33-s − 2·34-s + 36-s + 6·37-s − 38-s − 2·39-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 + 1.20·11-s − 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.852·22-s − 0.834·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s + 0.986·37-s − 0.162·38-s − 0.320·39-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: \(1\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.65180157442804, −13.09641790572453, −12.51080743208815, −12.11807254753784, −11.56518154503120, −11.35151724523740, −10.84051695849693, −10.03708747597108, −9.953256160039166, −9.397533871729135, −8.787978087406416, −8.255253649896686, −7.990442622302193, −7.046548210807646, −6.908036032343083, −6.230387031200699, −5.893211825553529, −5.277380882796938, −4.574384428941543, −4.013983466784827, −3.448730776839673, −2.854981395096310, −1.915634003046202, −1.426327449159688, −0.8661980898294735, 0, 0.8661980898294735, 1.426327449159688, 1.915634003046202, 2.854981395096310, 3.448730776839673, 4.013983466784827, 4.574384428941543, 5.277380882796938, 5.893211825553529, 6.230387031200699, 6.908036032343083, 7.046548210807646, 7.990442622302193, 8.255253649896686, 8.787978087406416, 9.397533871729135, 9.953256160039166, 10.03708747597108, 10.84051695849693, 11.35151724523740, 11.56518154503120, 12.11807254753784, 12.51080743208815, 13.09641790572453, 13.65180157442804

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