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Properties

Label 2-369600-1.1-c1-0-316
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 − 2·13-s − 2·17-s + 4·19-s − 21-s − 27-s − 6·29-s + 33-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s + 49-s + 2·51-s − 2·53-s − 4·57-s + 4·59-s − 6·61-s + 63-s − 12·67-s − 10·73-s − 77-s − 8·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 0.192·27-s − 1.11·29-s + 0.174·33-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.280·51-s − 0.274·53-s − 0.529·57-s + 0.520·59-s − 0.768·61-s + 0.125·63-s − 1.46·67-s − 1.17·73-s − 0.113·77-s − 0.900·79-s + 1/9·81-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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.79572698226762, −12.14243647089248, −11.68240365078183, −11.56529414910785, −10.89013865912569, −10.55066259562760, −10.10359229035393, −9.467281574728204, −9.287043917815050, −8.625403874230454, −8.097641371165562, −7.553746086771057, −7.273692348817583, −6.831388266915738, −6.021287962278638, −5.840130120173011, −5.260605990243599, −4.690445198030724, −4.462251381924596, −3.727097831541058, −3.155651149307359, −2.585209309006648, −1.934331955820796, −1.426750097847657, −0.6691721348222473, 0, 0.6691721348222473, 1.426750097847657, 1.934331955820796, 2.585209309006648, 3.155651149307359, 3.727097831541058, 4.462251381924596, 4.690445198030724, 5.260605990243599, 5.840130120173011, 6.021287962278638, 6.831388266915738, 7.273692348817583, 7.553746086771057, 8.097641371165562, 8.625403874230454, 9.287043917815050, 9.467281574728204, 10.10359229035393, 10.55066259562760, 10.89013865912569, 11.56529414910785, 11.68240365078183, 12.14243647089248, 12.79572698226762

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