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Properties

Label 2-177450-1.1-c1-0-111
Degree $2$
Conductor $177450$
Sign $1$
Analytic cond. $1416.94$
Root an. cond. $37.6423$
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 − 7-s + 8-s + 9-s − 11-s + 12-s − 14-s + 16-s + 6·17-s + 18-s + 19-s − 21-s − 22-s + 2·23-s + 24-s + 27-s − 28-s − 31-s + 32-s − 33-s + 6·34-s + 36-s + 38-s + 5·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.229·19-s − 0.218·21-s − 0.213·22-s + 0.417·23-s + 0.204·24-s + 0.192·27-s − 0.188·28-s − 0.179·31-s + 0.176·32-s − 0.174·33-s + 1.02·34-s + 1/6·36-s + 0.162·38-s + 0.780·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1416.94\)
Root analytic conductor: \(37.6423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.544353525\)
\(L(\frac12)\) \(\approx\) \(6.544353525\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 \)
good11 \( 1 + T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 17 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−13.15804139185812, −12.74350022848600, −12.41148321825001, −11.88364894932632, −11.36698973471671, −10.86890664386770, −10.20157284987009, −10.01550178862880, −9.367071344875007, −8.935154208611294, −8.286593728157396, −7.774242789083386, −7.418579368391253, −6.923858431260815, −6.244533938039067, −5.857837366786088, −5.181778045878503, −4.884811760251212, −4.103544878350431, −3.463753989739222, −3.344667549411951, −2.511957726572772, −2.136836387151450, −1.216684311145996, −0.6630553248092083, 0.6630553248092083, 1.216684311145996, 2.136836387151450, 2.511957726572772, 3.344667549411951, 3.463753989739222, 4.103544878350431, 4.884811760251212, 5.181778045878503, 5.857837366786088, 6.244533938039067, 6.923858431260815, 7.418579368391253, 7.774242789083386, 8.286593728157396, 8.935154208611294, 9.367071344875007, 10.01550178862880, 10.20157284987009, 10.86890664386770, 11.36698973471671, 11.88364894932632, 12.41148321825001, 12.74350022848600, 13.15804139185812

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