Properties

Label 2-2268-1.1-c1-0-4
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  − 1.73·5-s + 7-s − 3.46·11-s + 5·13-s − 1.73·17-s + 2·19-s + 3.46·23-s − 2.00·25-s − 1.73·29-s + 2·31-s − 1.73·35-s − 7·37-s + 6.92·41-s + 8·43-s − 10.3·47-s + 49-s + 13.8·53-s + 5.99·55-s + 10.3·59-s − 61-s − 8.66·65-s + 2·67-s + 11·73-s − 3.46·77-s + 14·79-s + 3.46·83-s + 2.99·85-s + ⋯
L(s)  = 1  − 0.774·5-s + 0.377·7-s − 1.04·11-s + 1.38·13-s − 0.420·17-s + 0.458·19-s + 0.722·23-s − 0.400·25-s − 0.321·29-s + 0.359·31-s − 0.292·35-s − 1.15·37-s + 1.08·41-s + 1.21·43-s − 1.51·47-s + 0.142·49-s + 1.90·53-s + 0.809·55-s + 1.35·59-s − 0.128·61-s − 1.07·65-s + 0.244·67-s + 1.28·73-s − 0.394·77-s + 1.57·79-s + 0.380·83-s + 0.325·85-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

\(L(1)\) \(\approx\) \(1.501453157\)
\(L(\frac12)\) \(\approx\) \(1.501453157\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 - 1.73T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.816874804479432557322232197721, −8.238281701572889519417981810853, −7.59942029419832757322927791745, −6.81137645569507189023563203258, −5.78980953826140060719453181803, −5.06505742123967975275650776555, −4.07985423647331204119384268824, −3.35448989415524127485545985783, −2.20187330442277585990298382092, −0.799407479757812069684742056716, 0.799407479757812069684742056716, 2.20187330442277585990298382092, 3.35448989415524127485545985783, 4.07985423647331204119384268824, 5.06505742123967975275650776555, 5.78980953826140060719453181803, 6.81137645569507189023563203258, 7.59942029419832757322927791745, 8.238281701572889519417981810853, 8.816874804479432557322232197721

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