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Properties

Label 2-32760-1.1-c1-0-20
Degree $2$
Conductor $32760$
Sign $-1$
Analytic cond. $261.589$
Root an. cond. $16.1737$
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  − 5-s − 7-s + 4·11-s − 13-s − 6·17-s − 6·23-s + 25-s + 6·29-s + 6·31-s + 35-s − 6·37-s + 2·41-s + 49-s − 4·55-s − 4·59-s + 2·61-s + 65-s + 2·67-s + 8·71-s − 8·73-s − 4·77-s + 12·83-s + 6·85-s + 14·89-s + 91-s − 8·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 1.20·11-s − 0.277·13-s − 1.45·17-s − 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.07·31-s + 0.169·35-s − 0.986·37-s + 0.312·41-s + 1/7·49-s − 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.124·65-s + 0.244·67-s + 0.949·71-s − 0.936·73-s − 0.455·77-s + 1.31·83-s + 0.650·85-s + 1.48·89-s + 0.104·91-s − 0.812·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32760\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(261.589\)
Root analytic conductor: \(16.1737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 32760,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−15.46734046033448, −14.72880582373910, −14.22954843524428, −13.72072049850103, −13.31467337030442, −12.43505761778014, −12.15253046352295, −11.68081725244124, −11.11358797150390, −10.46747462278418, −9.955251115297673, −9.341947769318558, −8.795050381647862, −8.352350370814504, −7.680764080131067, −6.965394700585408, −6.436146923951865, −6.193293943405707, −5.153623123035972, −4.494977130160313, −4.050767329058523, −3.412120464162857, −2.579158480625028, −1.903854416222239, −0.9352311594568969, 0, 0.9352311594568969, 1.903854416222239, 2.579158480625028, 3.412120464162857, 4.050767329058523, 4.494977130160313, 5.153623123035972, 6.193293943405707, 6.436146923951865, 6.965394700585408, 7.680764080131067, 8.352350370814504, 8.795050381647862, 9.341947769318558, 9.955251115297673, 10.46747462278418, 11.11358797150390, 11.68081725244124, 12.15253046352295, 12.43505761778014, 13.31467337030442, 13.72072049850103, 14.22954843524428, 14.72880582373910, 15.46734046033448

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