Properties

Label 2-369600-1.1-c1-0-11
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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  − 3-s − 7-s + 9-s − 11-s − 4·13-s + 5·17-s + 19-s + 21-s − 5·23-s − 27-s − 3·29-s + 6·31-s + 33-s − 12·37-s + 4·39-s − 2·41-s − 13·43-s − 6·47-s + 49-s − 5·51-s + 53-s − 57-s − 11·59-s − 5·61-s − 63-s + 10·67-s + 5·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 1.21·17-s + 0.229·19-s + 0.218·21-s − 1.04·23-s − 0.192·27-s − 0.557·29-s + 1.07·31-s + 0.174·33-s − 1.97·37-s + 0.640·39-s − 0.312·41-s − 1.98·43-s − 0.875·47-s + 1/7·49-s − 0.700·51-s + 0.137·53-s − 0.132·57-s − 1.43·59-s − 0.640·61-s − 0.125·63-s + 1.22·67-s + 0.601·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4467508745\)
\(L(\frac12)\) \(\approx\) \(0.4467508745\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−12.40241484359836, −11.96676451769686, −11.82508980620953, −11.24245312071343, −10.42672290681868, −10.32427909895336, −9.903682559241059, −9.471523351128231, −8.978116773780527, −8.154543622532400, −7.979623456835395, −7.476592310360350, −6.840111392159022, −6.557067216266242, −6.003563465579454, −5.381545808286375, −5.067744165624909, −4.708382924666463, −3.913519209699984, −3.341380734493395, −3.090265368592852, −2.139348424776582, −1.804705134467615, −1.007052634377693, −0.1960817715158567, 0.1960817715158567, 1.007052634377693, 1.804705134467615, 2.139348424776582, 3.090265368592852, 3.341380734493395, 3.913519209699984, 4.708382924666463, 5.067744165624909, 5.381545808286375, 6.003563465579454, 6.557067216266242, 6.840111392159022, 7.476592310360350, 7.979623456835395, 8.154543622532400, 8.978116773780527, 9.471523351128231, 9.903682559241059, 10.32427909895336, 10.42672290681868, 11.24245312071343, 11.82508980620953, 11.96676451769686, 12.40241484359836

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