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Properties

Label 2-139650-1.1-c1-0-129
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 + 4·13-s + 16-s − 2·17-s + 18-s + 19-s + 4·22-s + 24-s + 4·26-s + 27-s − 6·29-s + 32-s + 4·33-s − 2·34-s + 36-s + 8·37-s + 38-s + 4·39-s + 2·43-s + 4·44-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 + 1.10·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s + 0.852·22-s + 0.204·24-s + 0.784·26-s + 0.192·27-s − 1.11·29-s + 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s + 1.31·37-s + 0.162·38-s + 0.640·39-s + 0.304·43-s + 0.603·44-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

\(L(1)\) \(\approx\) \(7.924355536\)
\(L(\frac12)\) \(\approx\) \(7.924355536\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−13.49985098333613, −12.89583998695700, −12.72651582550367, −11.94324200958261, −11.53940531149225, −11.11442577091477, −10.73878400295397, −9.915445966913348, −9.593732631879266, −8.921838178751954, −8.651275769470073, −8.075128274848820, −7.325035314090343, −7.085201500425441, −6.394754365752770, −5.955532917391582, −5.533162503207837, −4.686349725234925, −4.193334057302750, −3.703982217430801, −3.433499158625887, −2.495253336064319, −2.101437361508045, −1.287734464407843, −0.7656589869483988, 0.7656589869483988, 1.287734464407843, 2.101437361508045, 2.495253336064319, 3.433499158625887, 3.703982217430801, 4.193334057302750, 4.686349725234925, 5.533162503207837, 5.955532917391582, 6.394754365752770, 7.085201500425441, 7.325035314090343, 8.075128274848820, 8.651275769470073, 8.921838178751954, 9.593732631879266, 9.915445966913348, 10.73878400295397, 11.11442577091477, 11.53940531149225, 11.94324200958261, 12.72651582550367, 12.89583998695700, 13.49985098333613

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