Properties

Label 2-215600-1.1-c1-0-108
Degree $2$
Conductor $215600$
Sign $-1$
Analytic cond. $1721.57$
Root an. cond. $41.4918$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 11-s + 2·13-s − 2·17-s − 4·19-s − 8·23-s − 2·29-s − 4·31-s − 6·37-s − 6·41-s − 4·43-s + 12·47-s + 10·53-s + 8·59-s − 10·61-s + 4·67-s + 8·71-s − 2·73-s + 8·79-s + 9·81-s − 12·83-s − 10·89-s + 10·97-s − 3·99-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 9-s + 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s − 0.371·29-s − 0.718·31-s − 0.986·37-s − 0.937·41-s − 0.609·43-s + 1.75·47-s + 1.37·53-s + 1.04·59-s − 1.28·61-s + 0.488·67-s + 0.949·71-s − 0.234·73-s + 0.900·79-s + 81-s − 1.31·83-s − 1.05·89-s + 1.01·97-s − 0.301·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(215600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(1721.57\)
Root analytic conductor: \(41.4918\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{215600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 215600,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−13.31773660442914, −12.73725168510355, −12.15901424404047, −11.90866925283002, −11.30998119210764, −10.99338319953260, −10.35498902901791, −10.12851388598536, −9.368415410627940, −8.852595432261435, −8.546496072063707, −8.218370019922677, −7.479436784599902, −7.038116648634233, −6.374390514732943, −6.068876570394360, −5.529011322299152, −5.082610098977956, −4.286270150577294, −3.827474881977762, −3.481204935914101, −2.638823398406406, −2.093225073880803, −1.676039537517895, −0.6395461924406758, 0, 0.6395461924406758, 1.676039537517895, 2.093225073880803, 2.638823398406406, 3.481204935914101, 3.827474881977762, 4.286270150577294, 5.082610098977956, 5.529011322299152, 6.068876570394360, 6.374390514732943, 7.038116648634233, 7.479436784599902, 8.218370019922677, 8.546496072063707, 8.852595432261435, 9.368415410627940, 10.12851388598536, 10.35498902901791, 10.99338319953260, 11.30998119210764, 11.90866925283002, 12.15901424404047, 12.73725168510355, 13.31773660442914

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