Properties

Label 2-369600-1.1-c1-0-125
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 + 6·17-s − 21-s − 2·23-s + 27-s − 8·29-s − 10·31-s + 33-s + 8·37-s + 4·39-s + 10·41-s + 4·43-s + 49-s + 6·51-s − 10·53-s − 2·61-s − 63-s + 16·67-s − 2·69-s − 8·71-s + 12·73-s − 77-s − 10·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 1.45·17-s − 0.218·21-s − 0.417·23-s + 0.192·27-s − 1.48·29-s − 1.79·31-s + 0.174·33-s + 1.31·37-s + 0.640·39-s + 1.56·41-s + 0.609·43-s + 1/7·49-s + 0.840·51-s − 1.37·53-s − 0.256·61-s − 0.125·63-s + 1.95·67-s − 0.240·69-s − 0.949·71-s + 1.40·73-s − 0.113·77-s − 1.12·79-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

\(L(1)\) \(\approx\) \(3.173427996\)
\(L(\frac12)\) \(\approx\) \(3.173427996\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.60892673432859, −12.20984283557977, −11.48387651809695, −11.03505224852696, −10.88354465137903, −10.15696242306310, −9.530366298266583, −9.421929697199321, −9.034822623506863, −8.285486499831837, −7.936438663586375, −7.553653418885375, −7.090704106663657, −6.428081672640444, −5.975929410066223, −5.571296444097880, −5.142600528050490, −4.160730699321236, −3.861145748913084, −3.612551092991621, −2.843200239803068, −2.460463683778717, −1.515681499201669, −1.340529159623187, −0.4421050774638833, 0.4421050774638833, 1.340529159623187, 1.515681499201669, 2.460463683778717, 2.843200239803068, 3.612551092991621, 3.861145748913084, 4.160730699321236, 5.142600528050490, 5.571296444097880, 5.975929410066223, 6.428081672640444, 7.090704106663657, 7.553653418885375, 7.936438663586375, 8.285486499831837, 9.034822623506863, 9.421929697199321, 9.530366298266583, 10.15696242306310, 10.88354465137903, 11.03505224852696, 11.48387651809695, 12.20984283557977, 12.60892673432859

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