Properties

Label 2-2184-1.1-c1-0-6
Degree $2$
Conductor $2184$
Sign $1$
Analytic cond. $17.4393$
Root an. cond. $4.17604$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 0.152·5-s − 7-s + 9-s + 0.385·11-s + 13-s + 0.152·15-s + 7.43·17-s − 7.20·19-s + 21-s + 2.90·23-s − 4.97·25-s − 27-s − 5.20·29-s + 1.76·31-s − 0.385·33-s + 0.152·35-s + 7.43·37-s − 39-s + 7.05·41-s + 2.90·43-s − 0.152·45-s − 3.59·47-s + 49-s − 7.43·51-s − 10.9·53-s − 0.0587·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.0681·5-s − 0.377·7-s + 0.333·9-s + 0.116·11-s + 0.277·13-s + 0.0393·15-s + 1.80·17-s − 1.65·19-s + 0.218·21-s + 0.604·23-s − 0.995·25-s − 0.192·27-s − 0.966·29-s + 0.317·31-s − 0.0670·33-s + 0.0257·35-s + 1.22·37-s − 0.160·39-s + 1.10·41-s + 0.442·43-s − 0.0227·45-s − 0.523·47-s + 0.142·49-s − 1.04·51-s − 1.50·53-s − 0.00791·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2184\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(17.4393\)
Root analytic conductor: \(4.17604\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2184,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.301449511\)
\(L(\frac12)\) \(\approx\) \(1.301449511\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 + 0.152T + 5T^{2} \)
11 \( 1 - 0.385T + 11T^{2} \)
17 \( 1 - 7.43T + 17T^{2} \)
19 \( 1 + 7.20T + 19T^{2} \)
23 \( 1 - 2.90T + 23T^{2} \)
29 \( 1 + 5.20T + 29T^{2} \)
31 \( 1 - 1.76T + 31T^{2} \)
37 \( 1 - 7.43T + 37T^{2} \)
41 \( 1 - 7.05T + 41T^{2} \)
43 \( 1 - 2.90T + 43T^{2} \)
47 \( 1 + 3.59T + 47T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 + 5.82T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 - 9.80T + 67T^{2} \)
71 \( 1 - 5.82T + 71T^{2} \)
73 \( 1 - 3.09T + 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + 5.59T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−9.258598089574687069947392102830, −8.102871384828789535100940132682, −7.61612924038700728879179582213, −6.50787008838141002125419353245, −6.01702274126565034885151224811, −5.17687241990234323073743359119, −4.15731785954268784274056877998, −3.39128726755421323402875748274, −2.10678331429427235314045644495, −0.77120399444964110859437206124, 0.77120399444964110859437206124, 2.10678331429427235314045644495, 3.39128726755421323402875748274, 4.15731785954268784274056877998, 5.17687241990234323073743359119, 6.01702274126565034885151224811, 6.50787008838141002125419353245, 7.61612924038700728879179582213, 8.102871384828789535100940132682, 9.258598089574687069947392102830

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