Properties

Label 2-397800-1.1-c1-0-3
Degree $2$
Conductor $397800$
Sign $1$
Analytic cond. $3176.44$
Root an. cond. $56.3599$
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  − 2·11-s + 13-s − 17-s − 8·19-s + 6·23-s − 4·31-s − 2·37-s − 4·41-s + 4·43-s − 7·49-s − 6·53-s − 4·59-s + 8·61-s − 8·67-s + 4·71-s + 10·79-s + 8·83-s − 6·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.603·11-s + 0.277·13-s − 0.242·17-s − 1.83·19-s + 1.25·23-s − 0.718·31-s − 0.328·37-s − 0.624·41-s + 0.609·43-s − 49-s − 0.824·53-s − 0.520·59-s + 1.02·61-s − 0.977·67-s + 0.474·71-s + 1.12·79-s + 0.878·83-s − 0.635·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(397800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(3176.44\)
Root analytic conductor: \(56.3599\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{397800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 397800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6425360423\)
\(L(\frac12)\) \(\approx\) \(0.6425360423\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−12.61551110204013, −12.05397554952454, −11.35784968666071, −11.06823438418467, −10.69390486888237, −10.30559627048307, −9.760561533689311, −9.139994076909742, −8.866573831754931, −8.367094233575685, −7.946946008463871, −7.397223229742987, −6.923418473513332, −6.368579517366407, −6.135406812322553, −5.387735395484326, −4.891346216311190, −4.630402425562240, −3.795622394159432, −3.558335844906783, −2.742799523538620, −2.366991813238423, −1.727751137869027, −1.133568305087184, −0.2073170442129504, 0.2073170442129504, 1.133568305087184, 1.727751137869027, 2.366991813238423, 2.742799523538620, 3.558335844906783, 3.795622394159432, 4.630402425562240, 4.891346216311190, 5.387735395484326, 6.135406812322553, 6.368579517366407, 6.923418473513332, 7.397223229742987, 7.946946008463871, 8.367094233575685, 8.866573831754931, 9.139994076909742, 9.760561533689311, 10.30559627048307, 10.69390486888237, 11.06823438418467, 11.35784968666071, 12.05397554952454, 12.61551110204013

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