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Properties

Label 2-35280-1.1-c1-0-8
Degree $2$
Conductor $35280$
Sign $1$
Analytic cond. $281.712$
Root an. cond. $16.7842$
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  − 5-s − 2·13-s − 6·17-s − 4·19-s + 25-s + 6·29-s − 4·31-s + 2·37-s + 6·41-s − 8·43-s + 12·47-s − 6·53-s + 12·59-s − 2·61-s + 2·65-s − 8·67-s − 14·73-s + 16·79-s − 12·83-s + 6·85-s + 6·89-s + 4·95-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1/5·25-s + 1.11·29-s − 0.718·31-s + 0.328·37-s + 0.937·41-s − 1.21·43-s + 1.75·47-s − 0.824·53-s + 1.56·59-s − 0.256·61-s + 0.248·65-s − 0.977·67-s − 1.63·73-s + 1.80·79-s − 1.31·83-s + 0.650·85-s + 0.635·89-s + 0.410·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35280\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(281.712\)
Root analytic conductor: \(16.7842\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 35280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9566543179\)
\(L(\frac12)\) \(\approx\) \(0.9566543179\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.96747786159393, −14.52342349997826, −13.88646792304335, −13.29853252495068, −12.86463085109171, −12.30132087308224, −11.80696656619429, −11.21742990847035, −10.71169795402294, −10.29243180186000, −9.537643977720484, −8.949361164603703, −8.553845103703815, −7.926796150668850, −7.289914256852668, −6.756998050656486, −6.271552618252363, −5.514997003285332, −4.776571214839052, −4.296471265113570, −3.789424381353523, −2.760258966550732, −2.372696675344851, −1.447523101929682, −0.3615469832874573, 0.3615469832874573, 1.447523101929682, 2.372696675344851, 2.760258966550732, 3.789424381353523, 4.296471265113570, 4.776571214839052, 5.514997003285332, 6.271552618252363, 6.756998050656486, 7.289914256852668, 7.926796150668850, 8.553845103703815, 8.949361164603703, 9.537643977720484, 10.29243180186000, 10.71169795402294, 11.21742990847035, 11.80696656619429, 12.30132087308224, 12.86463085109171, 13.29853252495068, 13.88646792304335, 14.52342349997826, 14.96747786159393

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