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Properties

Label 2-369600-1.1-c1-0-146
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 − 2·13-s + 6·17-s + 4·19-s − 21-s − 8·23-s − 27-s − 6·29-s + 8·31-s + 33-s + 6·37-s + 2·39-s − 6·41-s − 4·43-s + 8·47-s + 49-s − 6·51-s − 10·53-s − 4·57-s + 12·59-s + 10·61-s + 63-s − 12·67-s + 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.218·21-s − 1.66·23-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.174·33-s + 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.840·51-s − 1.37·53-s − 0.529·57-s + 1.56·59-s + 1.28·61-s + 0.125·63-s − 1.46·67-s + 0.963·69-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: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.251473513\)
\(L(\frac12)\) \(\approx\) \(2.251473513\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−12.24477109772763, −11.99820806624895, −11.76794598998931, −11.23809076653135, −10.67320305308172, −10.11636743532819, −9.907709925679539, −9.559493308396687, −8.888451566627969, −8.160217487458868, −7.894216530837906, −7.582351851204717, −6.974949540482210, −6.459794645970644, −5.852186331773709, −5.467737277572825, −5.205617436415013, −4.424132753004267, −4.152704327838406, −3.352827899041464, −3.005104905046828, −2.163488822974038, −1.749308730851009, −0.9643802243065671, −0.4712146231319109, 0.4712146231319109, 0.9643802243065671, 1.749308730851009, 2.163488822974038, 3.005104905046828, 3.352827899041464, 4.152704327838406, 4.424132753004267, 5.205617436415013, 5.467737277572825, 5.852186331773709, 6.459794645970644, 6.974949540482210, 7.582351851204717, 7.894216530837906, 8.160217487458868, 8.888451566627969, 9.559493308396687, 9.907709925679539, 10.11636743532819, 10.67320305308172, 11.23809076653135, 11.76794598998931, 11.99820806624895, 12.24477109772763

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