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Properties

Label 2-139650-1.1-c1-0-119
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 + 4·11-s + 12-s + 6·13-s + 16-s + 2·17-s − 18-s + 19-s − 4·22-s + 8·23-s − 24-s − 6·26-s + 27-s − 2·29-s − 8·31-s − 32-s + 4·33-s − 2·34-s + 36-s + 10·37-s − 38-s + 6·39-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 + 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.852·22-s + 1.66·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s + 1.64·37-s − 0.162·38-s + 0.960·39-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

\(L(1)\) \(\approx\) \(3.863425081\)
\(L(\frac12)\) \(\approx\) \(3.863425081\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.37629965668011, −13.02266184011744, −12.51409435796399, −11.75715962322205, −11.41423192930911, −11.04312528640500, −10.54455694382132, −9.871026180876437, −9.389994762944380, −9.094879543741614, −8.608280499853364, −8.147101944035765, −7.711384652543974, −6.897618242078089, −6.694758242560937, −6.216373549918665, −5.353676545253547, −5.063218932957665, −3.925250103677772, −3.684140912976985, −3.284419835643269, −2.421061129817910, −1.750732044951235, −1.153848710040711, −0.7339228008039694, 0.7339228008039694, 1.153848710040711, 1.750732044951235, 2.421061129817910, 3.284419835643269, 3.684140912976985, 3.925250103677772, 5.063218932957665, 5.353676545253547, 6.216373549918665, 6.694758242560937, 6.897618242078089, 7.711384652543974, 8.147101944035765, 8.608280499853364, 9.094879543741614, 9.389994762944380, 9.871026180876437, 10.54455694382132, 11.04312528640500, 11.41423192930911, 11.75715962322205, 12.51409435796399, 13.02266184011744, 13.37629965668011

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