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Properties

Label 2-10800-1.1-c1-0-65
Degree $2$
Conductor $10800$
Sign $-1$
Analytic cond. $86.2384$
Root an. cond. $9.28646$
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  − 3·7-s + 4·11-s − 13-s − 4·17-s + 19-s − 4·23-s + 4·31-s + 9·37-s − 8·43-s + 12·47-s + 2·49-s − 8·53-s + 4·59-s − 5·61-s + 11·67-s + 8·71-s − 73-s − 12·77-s + 5·79-s − 8·83-s − 12·89-s + 3·91-s − 5·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.13·7-s + 1.20·11-s − 0.277·13-s − 0.970·17-s + 0.229·19-s − 0.834·23-s + 0.718·31-s + 1.47·37-s − 1.21·43-s + 1.75·47-s + 2/7·49-s − 1.09·53-s + 0.520·59-s − 0.640·61-s + 1.34·67-s + 0.949·71-s − 0.117·73-s − 1.36·77-s + 0.562·79-s − 0.878·83-s − 1.27·89-s + 0.314·91-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10800\)    =    \(2^{4} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(86.2384\)
Root analytic conductor: \(9.28646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−16.71991713262004, −16.35720760997299, −15.55238580966118, −15.32159282759827, −14.43625049427264, −13.95934320949199, −13.40657726008287, −12.74340814354712, −12.27813226365406, −11.61419960368891, −11.12930982023605, −10.27943493996147, −9.636914914021085, −9.375659798389157, −8.609720573114505, −7.938288424120973, −7.081251288486498, −6.480897828176258, −6.193005243346449, −5.254422121491403, −4.294773500416454, −3.872475100643728, −2.974269200278088, −2.237138072717003, −1.131423639064996, 0, 1.131423639064996, 2.237138072717003, 2.974269200278088, 3.872475100643728, 4.294773500416454, 5.254422121491403, 6.193005243346449, 6.480897828176258, 7.081251288486498, 7.938288424120973, 8.609720573114505, 9.375659798389157, 9.636914914021085, 10.27943493996147, 11.12930982023605, 11.61419960368891, 12.27813226365406, 12.74340814354712, 13.40657726008287, 13.95934320949199, 14.43625049427264, 15.32159282759827, 15.55238580966118, 16.35720760997299, 16.71991713262004

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