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Properties

Label 2-143570-1.1-c1-0-9
Degree $2$
Conductor $143570$
Sign $1$
Analytic cond. $1146.41$
Root an. cond. $33.8587$
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 + 3·3-s + 4-s − 5-s + 3·6-s + 8-s + 6·9-s − 10-s + 5·11-s + 3·12-s − 7·13-s − 3·15-s + 16-s + 3·17-s + 6·18-s + 8·19-s − 20-s + 5·22-s − 4·23-s + 3·24-s + 25-s − 7·26-s + 9·27-s + 2·29-s − 3·30-s + 10·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s + 0.353·8-s + 2·9-s − 0.316·10-s + 1.50·11-s + 0.866·12-s − 1.94·13-s − 0.774·15-s + 1/4·16-s + 0.727·17-s + 1.41·18-s + 1.83·19-s − 0.223·20-s + 1.06·22-s − 0.834·23-s + 0.612·24-s + 1/5·25-s − 1.37·26-s + 1.73·27-s + 0.371·29-s − 0.547·30-s + 1.79·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143570\)    =    \(2 \cdot 5 \cdot 7^{2} \cdot 293\)
Sign: $1$
Analytic conductor: \(1146.41\)
Root analytic conductor: \(33.8587\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 143570,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(10.58074268\)
\(L(\frac12)\) \(\approx\) \(10.58074268\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
293 \( 1 - T \)
good3 \( 1 - p T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.82535128598885, −12.81873157061912, −12.45536896657047, −12.08298988593849, −11.77785925857504, −11.16519277456093, −10.23436961414837, −9.893768393348684, −9.443134259385383, −9.238968944276902, −8.325520509492375, −7.989267909641812, −7.591014560683565, −6.981793223795787, −6.809540258865177, −5.914597132456794, −5.227998852411961, −4.586392751642127, −4.289231579293846, −3.524349377860345, −3.268667442449812, −2.701602695433645, −2.136666634024163, −1.439657609075877, −0.7776609194589745, 0.7776609194589745, 1.439657609075877, 2.136666634024163, 2.701602695433645, 3.268667442449812, 3.524349377860345, 4.289231579293846, 4.586392751642127, 5.227998852411961, 5.914597132456794, 6.809540258865177, 6.981793223795787, 7.591014560683565, 7.989267909641812, 8.325520509492375, 9.238968944276902, 9.443134259385383, 9.893768393348684, 10.23436961414837, 11.16519277456093, 11.77785925857504, 12.08298988593849, 12.45536896657047, 12.81873157061912, 13.82535128598885

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