Properties

Label 2-369600-1.1-c1-0-1
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 − 6·13-s + 7·17-s + 5·19-s + 21-s − 3·23-s − 27-s − 5·29-s − 4·31-s + 33-s − 4·37-s + 6·39-s + 2·41-s − 5·43-s − 6·47-s + 49-s − 7·51-s − 11·53-s − 5·57-s − 59-s − 3·61-s − 63-s − 16·67-s + 3·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 1.69·17-s + 1.14·19-s + 0.218·21-s − 0.625·23-s − 0.192·27-s − 0.928·29-s − 0.718·31-s + 0.174·33-s − 0.657·37-s + 0.960·39-s + 0.312·41-s − 0.762·43-s − 0.875·47-s + 1/7·49-s − 0.980·51-s − 1.51·53-s − 0.662·57-s − 0.130·59-s − 0.384·61-s − 0.125·63-s − 1.95·67-s + 0.361·69-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

\(L(1)\) \(\approx\) \(0.2084477898\)
\(L(\frac12)\) \(\approx\) \(0.2084477898\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−12.39321831648827, −12.06323021388504, −11.75319034411827, −11.22007974390167, −10.55955184530452, −10.27584104143693, −9.766628285420874, −9.410999960122283, −9.194377121229101, −8.146254063764897, −7.754677249599150, −7.575987518941466, −6.996505169588769, −6.507904566237792, −5.865347642557068, −5.494180477123048, −4.965031567515492, −4.822304632478343, −3.829371025956181, −3.450932144875462, −2.977232589312946, −2.290180678987324, −1.654108280438769, −1.077120722142449, −0.1299432269639483, 0.1299432269639483, 1.077120722142449, 1.654108280438769, 2.290180678987324, 2.977232589312946, 3.450932144875462, 3.829371025956181, 4.822304632478343, 4.965031567515492, 5.494180477123048, 5.865347642557068, 6.507904566237792, 6.996505169588769, 7.575987518941466, 7.754677249599150, 8.146254063764897, 9.194377121229101, 9.410999960122283, 9.766628285420874, 10.27584104143693, 10.55955184530452, 11.22007974390167, 11.75319034411827, 12.06323021388504, 12.39321831648827

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