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Properties

Label 2-139650-1.1-c1-0-142
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 + 5·13-s + 16-s + 7·17-s − 18-s − 19-s + 4·22-s − 4·23-s + 24-s − 5·26-s − 27-s + 7·29-s − 3·31-s − 32-s + 4·33-s − 7·34-s + 36-s − 9·37-s + 38-s − 5·39-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 + 1.38·13-s + 1/4·16-s + 1.69·17-s − 0.235·18-s − 0.229·19-s + 0.852·22-s − 0.834·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s + 1.29·29-s − 0.538·31-s − 0.176·32-s + 0.696·33-s − 1.20·34-s + 1/6·36-s − 1.47·37-s + 0.162·38-s − 0.800·39-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 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−13.75583257457752, −12.92765506426715, −12.71668279973755, −12.02248710747935, −11.84252484059264, −11.01137082051380, −10.78199677272200, −10.24215291490873, −10.00695173624548, −9.385997346592047, −8.633591836655942, −8.325309459018191, −7.871259362886682, −7.352246530687647, −6.810735361342573, −6.127190464329416, −5.825516862327265, −5.286807415597107, −4.768498168340070, −3.886256515871543, −3.429698863665662, −2.833000406885557, −2.040934423579290, −1.387474820318725, −0.7992847941982631, 0, 0.7992847941982631, 1.387474820318725, 2.040934423579290, 2.833000406885557, 3.429698863665662, 3.886256515871543, 4.768498168340070, 5.286807415597107, 5.825516862327265, 6.127190464329416, 6.810735361342573, 7.352246530687647, 7.871259362886682, 8.325309459018191, 8.633591836655942, 9.385997346592047, 10.00695173624548, 10.24215291490873, 10.78199677272200, 11.01137082051380, 11.84252484059264, 12.02248710747935, 12.71668279973755, 12.92765506426715, 13.75583257457752

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