Properties

Label 2-2268-1.1-c1-0-7
Degree $2$
Conductor $2268$
Sign $1$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
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  + 1.11·5-s + 7-s + 1.88·11-s − 13-s + 5.87·17-s + 7.09·19-s − 3.98·23-s − 3.76·25-s − 0.987·29-s − 0.666·31-s + 1.11·35-s − 1.33·37-s + 1.88·41-s − 10.8·43-s + 11.0·47-s + 49-s + 12.2·53-s + 2.09·55-s − 4.76·59-s + 3.76·61-s − 1.11·65-s + 4.09·67-s + 10.7·71-s − 3.09·73-s + 1.88·77-s + 6.43·79-s + 11.8·83-s + ⋯
L(s)  = 1  + 0.496·5-s + 0.377·7-s + 0.569·11-s − 0.277·13-s + 1.42·17-s + 1.62·19-s − 0.831·23-s − 0.753·25-s − 0.183·29-s − 0.119·31-s + 0.187·35-s − 0.219·37-s + 0.294·41-s − 1.65·43-s + 1.61·47-s + 0.142·49-s + 1.67·53-s + 0.283·55-s − 0.620·59-s + 0.482·61-s − 0.137·65-s + 0.500·67-s + 1.27·71-s − 0.362·73-s + 0.215·77-s + 0.723·79-s + 1.30·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $1$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.253413794\)
\(L(\frac12)\) \(\approx\) \(2.253413794\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 1.11T + 5T^{2} \)
11 \( 1 - 1.88T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 5.87T + 17T^{2} \)
19 \( 1 - 7.09T + 19T^{2} \)
23 \( 1 + 3.98T + 23T^{2} \)
29 \( 1 + 0.987T + 29T^{2} \)
31 \( 1 + 0.666T + 31T^{2} \)
37 \( 1 + 1.33T + 37T^{2} \)
41 \( 1 - 1.88T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + 4.76T + 59T^{2} \)
61 \( 1 - 3.76T + 61T^{2} \)
67 \( 1 - 4.09T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 3.09T + 73T^{2} \)
79 \( 1 - 6.43T + 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 - 0.765T + 97T^{2} \)
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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

−9.170253474368430120861992693460, −8.137358786635108106077005880484, −7.55990863272468300779993836038, −6.73661088880707328437725493218, −5.62765808299818400022755349188, −5.35072332053585461880387847954, −4.08227221565028679776404096391, −3.27460322272164033596761099256, −2.07474786433568748434270171766, −1.04483332189833138687378324111, 1.04483332189833138687378324111, 2.07474786433568748434270171766, 3.27460322272164033596761099256, 4.08227221565028679776404096391, 5.35072332053585461880387847954, 5.62765808299818400022755349188, 6.73661088880707328437725493218, 7.55990863272468300779993836038, 8.137358786635108106077005880484, 9.170253474368430120861992693460

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