Properties

Label 2-139650-1.1-c1-0-139
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 − 2·11-s − 12-s − 2·13-s + 16-s + 2·17-s − 18-s + 19-s + 2·22-s + 6·23-s + 24-s + 2·26-s − 27-s − 6·29-s − 32-s + 2·33-s − 2·34-s + 36-s + 10·37-s − 38-s + 2·39-s + 8·41-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 − 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.229·19-s + 0.426·22-s + 1.25·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 1.11·29-s − 0.176·32-s + 0.348·33-s − 0.342·34-s + 1/6·36-s + 1.64·37-s − 0.162·38-s + 0.320·39-s + 1.24·41-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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.50552703753947, −12.96834957603755, −12.73146053217386, −12.15320394862699, −11.60220219574171, −11.17045819861005, −10.78926498415163, −10.35944768095773, −9.640124835100187, −9.463876698401173, −8.949833833177448, −8.184928021984934, −7.682517534907058, −7.450627545450196, −6.826619581855496, −6.277687855789875, −5.676704688306711, −5.291976720057084, −4.707579787258383, −4.072504193702117, −3.339832592254421, −2.690136796970209, −2.228506251500743, −1.318165210308901, −0.7896354106802817, 0, 0.7896354106802817, 1.318165210308901, 2.228506251500743, 2.690136796970209, 3.339832592254421, 4.072504193702117, 4.707579787258383, 5.291976720057084, 5.676704688306711, 6.277687855789875, 6.826619581855496, 7.450627545450196, 7.682517534907058, 8.184928021984934, 8.949833833177448, 9.463876698401173, 9.640124835100187, 10.35944768095773, 10.78926498415163, 11.17045819861005, 11.60220219574171, 12.15320394862699, 12.73146053217386, 12.96834957603755, 13.50552703753947

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