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Properties

Label 2-369600-1.1-c1-0-526
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 + 4·23-s − 27-s + 6·29-s − 33-s + 2·37-s − 2·39-s + 6·41-s + 12·43-s − 8·47-s + 49-s + 2·51-s − 6·53-s + 4·57-s − 8·59-s + 14·61-s + 63-s + 12·67-s − 4·69-s + 8·71-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 + 0.834·23-s − 0.192·27-s + 1.11·29-s − 0.174·33-s + 0.328·37-s − 0.320·39-s + 0.937·41-s + 1.82·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.529·57-s − 1.04·59-s + 1.79·61-s + 0.125·63-s + 1.46·67-s − 0.481·69-s + 0.949·71-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 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.72153209686485, −12.30092624956209, −11.77541728341891, −11.25780751739104, −10.93409888083402, −10.69669454231040, −10.11189027253732, −9.429029042247982, −9.223069664015681, −8.592384954046862, −8.159037873393948, −7.763216982710520, −7.087865580881147, −6.611144456738609, −6.320666320903271, −5.819977711951844, −5.209704804539022, −4.710374523339077, −4.350635072513977, −3.796715131654947, −3.229864567735543, −2.419607225813630, −2.120233502067267, −1.150281405222548, −0.9156081003089102, 0, 0.9156081003089102, 1.150281405222548, 2.120233502067267, 2.419607225813630, 3.229864567735543, 3.796715131654947, 4.350635072513977, 4.710374523339077, 5.209704804539022, 5.819977711951844, 6.320666320903271, 6.611144456738609, 7.087865580881147, 7.763216982710520, 8.159037873393948, 8.592384954046862, 9.223069664015681, 9.429029042247982, 10.11189027253732, 10.69669454231040, 10.93409888083402, 11.25780751739104, 11.77541728341891, 12.30092624956209, 12.72153209686485

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