Properties

Label 2-58800-1.1-c1-0-192
Degree $2$
Conductor $58800$
Sign $-1$
Analytic cond. $469.520$
Root an. cond. $21.6684$
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-s + 9-s + 13-s − 6·17-s − 5·19-s + 6·23-s + 27-s − 6·29-s − 5·31-s + 7·37-s + 39-s + 12·41-s − 43-s + 6·47-s − 6·51-s − 5·57-s + 6·59-s + 2·61-s − 7·67-s + 6·69-s − 12·71-s − 11·73-s + 13·79-s + 81-s − 12·83-s − 6·87-s + 6·89-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 0.277·13-s − 1.45·17-s − 1.14·19-s + 1.25·23-s + 0.192·27-s − 1.11·29-s − 0.898·31-s + 1.15·37-s + 0.160·39-s + 1.87·41-s − 0.152·43-s + 0.875·47-s − 0.840·51-s − 0.662·57-s + 0.781·59-s + 0.256·61-s − 0.855·67-s + 0.722·69-s − 1.42·71-s − 1.28·73-s + 1.46·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58800\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(469.520\)
Root analytic conductor: \(21.6684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.79003831413870, −14.19382858088299, −13.29647522493350, −13.14254513433307, −12.88321556787922, −12.11432588144928, −11.37694605901413, −10.96290096408654, −10.68555895835646, −9.922010454282218, −9.204581934921366, −8.940890540394544, −8.605564834312902, −7.708956697810334, −7.392556258962761, −6.770019871595133, −6.166767369641604, −5.663075369360187, −4.827524718818926, −4.233347645061875, −3.908072464205827, −3.000659918184536, −2.426121577680317, −1.892329863923324, −0.9898331118794423, 0, 0.9898331118794423, 1.892329863923324, 2.426121577680317, 3.000659918184536, 3.908072464205827, 4.233347645061875, 4.827524718818926, 5.663075369360187, 6.166767369641604, 6.770019871595133, 7.392556258962761, 7.708956697810334, 8.605564834312902, 8.940890540394544, 9.204581934921366, 9.922010454282218, 10.68555895835646, 10.96290096408654, 11.37694605901413, 12.11432588144928, 12.88321556787922, 13.14254513433307, 13.29647522493350, 14.19382858088299, 14.79003831413870

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