Properties

Label 2-139650-1.1-c1-0-128
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 − 6·11-s + 12-s + 13-s + 16-s + 2·17-s − 18-s − 19-s + 6·22-s − 23-s − 24-s − 26-s + 27-s − 9·29-s − 2·31-s − 32-s − 6·33-s − 2·34-s + 36-s + 10·37-s + 38-s + 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 − 1.80·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s + 1.27·22-s − 0.208·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 1.67·29-s − 0.359·31-s − 0.176·32-s − 1.04·33-s − 0.342·34-s + 1/6·36-s + 1.64·37-s + 0.162·38-s + 0.160·39-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−13.55805867609173, −13.10020362469025, −12.80834432769950, −12.31545613070480, −11.58990923275808, −11.06492511481176, −10.77213572333029, −10.16959238435254, −9.803898487531253, −9.314769019961112, −8.771478117158685, −8.243098955995817, −7.848671645926810, −7.423097729734987, −7.098335431944313, −6.147768130352053, −5.780033638596675, −5.267517374121834, −4.573508497799537, −3.905679365612714, −3.308452261715565, −2.683248803957509, −2.264729833269847, −1.628771979883575, −0.7624753647897921, 0, 0.7624753647897921, 1.628771979883575, 2.264729833269847, 2.683248803957509, 3.308452261715565, 3.905679365612714, 4.573508497799537, 5.267517374121834, 5.780033638596675, 6.147768130352053, 7.098335431944313, 7.423097729734987, 7.848671645926810, 8.243098955995817, 8.771478117158685, 9.314769019961112, 9.803898487531253, 10.16959238435254, 10.77213572333029, 11.06492511481176, 11.58990923275808, 12.31545613070480, 12.80834432769950, 13.10020362469025, 13.55805867609173

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