Properties

Label 2-35280-1.1-c1-0-1
Degree $2$
Conductor $35280$
Sign $1$
Analytic cond. $281.712$
Root an. cond. $16.7842$
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  − 5-s − 3·11-s − 5·13-s + 3·17-s + 2·19-s − 6·23-s + 25-s − 3·29-s − 4·31-s + 2·37-s − 12·41-s + 10·43-s − 9·47-s − 12·53-s + 3·55-s − 8·61-s + 5·65-s + 4·67-s − 2·73-s + 79-s − 12·83-s − 3·85-s − 12·89-s − 2·95-s + 97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.904·11-s − 1.38·13-s + 0.727·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s − 0.557·29-s − 0.718·31-s + 0.328·37-s − 1.87·41-s + 1.52·43-s − 1.31·47-s − 1.64·53-s + 0.404·55-s − 1.02·61-s + 0.620·65-s + 0.488·67-s − 0.234·73-s + 0.112·79-s − 1.31·83-s − 0.325·85-s − 1.27·89-s − 0.205·95-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35280\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(281.712\)
Root analytic conductor: \(16.7842\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{35280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 35280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4811801098\)
\(L(\frac12)\) \(\approx\) \(0.4811801098\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.86463551166346, −14.46322274158138, −14.00034809109534, −13.33569752151227, −12.70243325569366, −12.35046972998195, −11.86247311413565, −11.25838706731583, −10.72907678605715, −9.960051805238617, −9.807721517102572, −9.137422516059249, −8.226246313967079, −7.935756247690152, −7.388763057692942, −6.946064308439985, −6.017089968072308, −5.514750956076624, −4.896418317197660, −4.396885343902758, −3.503112598900266, −3.017702931367377, −2.231580549033484, −1.514172170395125, −0.2496855317920755, 0.2496855317920755, 1.514172170395125, 2.231580549033484, 3.017702931367377, 3.503112598900266, 4.396885343902758, 4.896418317197660, 5.514750956076624, 6.017089968072308, 6.946064308439985, 7.388763057692942, 7.935756247690152, 8.226246313967079, 9.137422516059249, 9.807721517102572, 9.960051805238617, 10.72907678605715, 11.25838706731583, 11.86247311413565, 12.35046972998195, 12.70243325569366, 13.33569752151227, 14.00034809109534, 14.46322274158138, 14.86463551166346

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