Properties

Label 2-140790-1.1-c1-0-48
Degree $2$
Conductor $140790$
Sign $1$
Analytic cond. $1124.21$
Root an. cond. $33.5292$
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 − 5-s − 6-s + 2·7-s − 8-s + 9-s + 10-s + 4·11-s + 12-s + 13-s − 2·14-s − 15-s + 16-s + 8·17-s − 18-s − 20-s + 2·21-s − 4·22-s + 6·23-s − 24-s + 25-s − 26-s + 27-s + 2·28-s + 4·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.94·17-s − 0.235·18-s − 0.223·20-s + 0.436·21-s − 0.852·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.377·28-s + 0.742·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140790\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1124.21\)
Root analytic conductor: \(33.5292\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{140790} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 140790,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.634805643\)
\(L(\frac12)\) \(\approx\) \(3.634805643\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 - T \)
19 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−13.46264455313928, −12.90679954190904, −12.18331014679868, −11.88037832522698, −11.66713531313155, −10.77138496134122, −10.64871845633071, −9.937899738876794, −9.432610753526800, −8.997824583604285, −8.578054481994376, −8.060858632533863, −7.586368805823985, −7.309524855414605, −6.553900394117755, −6.171577082085859, −5.367707385047700, −4.874644452670104, −4.199427332157511, −3.616962304879712, −3.123587039452195, −2.588847340572221, −1.530086930304891, −1.324202554777468, −0.6804130288416537, 0.6804130288416537, 1.324202554777468, 1.530086930304891, 2.588847340572221, 3.123587039452195, 3.616962304879712, 4.199427332157511, 4.874644452670104, 5.367707385047700, 6.171577082085859, 6.553900394117755, 7.309524855414605, 7.586368805823985, 8.060858632533863, 8.578054481994376, 8.997824583604285, 9.432610753526800, 9.937899738876794, 10.64871845633071, 10.77138496134122, 11.66713531313155, 11.88037832522698, 12.18331014679868, 12.90679954190904, 13.46264455313928

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