Properties

Label 2-369600-1.1-c1-0-334
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s − 11-s − 6·13-s + 2·17-s + 8·19-s + 21-s + 4·23-s − 27-s − 2·29-s − 8·31-s + 33-s + 6·37-s + 6·39-s − 2·41-s + 8·43-s + 4·47-s + 49-s − 2·51-s + 2·53-s − 8·57-s + 12·59-s − 10·61-s − 63-s − 12·67-s − 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.485·17-s + 1.83·19-s + 0.218·21-s + 0.834·23-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.174·33-s + 0.986·37-s + 0.960·39-s − 0.312·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.274·53-s − 1.05·57-s + 1.56·59-s − 1.28·61-s − 0.125·63-s − 1.46·67-s − 0.481·69-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.72025609598187, −12.14384734501433, −11.85667014997716, −11.53959437340626, −10.88594872526983, −10.40823584696006, −10.09854358057334, −9.545699663027192, −9.155738612739883, −8.922767457796704, −7.820553450471296, −7.605055361334141, −7.255357177748037, −6.928852852829962, −6.067872040190286, −5.694556030855364, −5.303252127071613, −4.859746253714115, −4.342938034773343, −3.686763536402856, −3.047776493394561, −2.732343541804764, −2.026946320764803, −1.282160957065272, −0.6847753455875377, 0, 0.6847753455875377, 1.282160957065272, 2.026946320764803, 2.732343541804764, 3.047776493394561, 3.686763536402856, 4.342938034773343, 4.859746253714115, 5.303252127071613, 5.694556030855364, 6.067872040190286, 6.928852852829962, 7.255357177748037, 7.605055361334141, 7.820553450471296, 8.922767457796704, 9.155738612739883, 9.545699663027192, 10.09854358057334, 10.40823584696006, 10.88594872526983, 11.53959437340626, 11.85667014997716, 12.14384734501433, 12.72025609598187

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