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Properties

Label 2-92400-1.1-c1-0-117
Degree $2$
Conductor $92400$
Sign $-1$
Analytic cond. $737.817$
Root an. cond. $27.1628$
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 − 6·17-s − 8·19-s − 21-s + 27-s − 2·29-s + 4·31-s − 33-s + 2·37-s + 2·39-s − 2·41-s + 4·43-s − 4·47-s + 49-s − 6·51-s − 14·53-s − 8·57-s − 4·59-s + 14·61-s − 63-s + 12·67-s + 8·71-s + 2·73-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 − 1.83·19-s − 0.218·21-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s − 0.840·51-s − 1.92·53-s − 1.05·57-s − 0.520·59-s + 1.79·61-s − 0.125·63-s + 1.46·67-s + 0.949·71-s + 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(737.817\)
Root analytic conductor: \(27.1628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ -1)\)

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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.08700705629232, −13.40967169074696, −13.23918775480356, −12.66961349070436, −12.38076874965399, −11.49095309323310, −10.97990389622323, −10.78102806934111, −10.07260435745175, −9.540970044054380, −9.077629773650247, −8.544250505107370, −8.159184754875659, −7.700207502896250, −6.756843349950235, −6.554475540446155, −6.147056033024860, −5.240023898263820, −4.671004993447619, −4.118259497861631, −3.647809538410280, −2.931816976989120, −2.188110989109066, −1.969912856388450, −0.8332698043158769, 0, 0.8332698043158769, 1.969912856388450, 2.188110989109066, 2.931816976989120, 3.647809538410280, 4.118259497861631, 4.671004993447619, 5.240023898263820, 6.147056033024860, 6.554475540446155, 6.756843349950235, 7.700207502896250, 8.159184754875659, 8.544250505107370, 9.077629773650247, 9.540970044054380, 10.07260435745175, 10.78102806934111, 10.97990389622323, 11.49095309323310, 12.38076874965399, 12.66961349070436, 13.23918775480356, 13.40967169074696, 14.08700705629232

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