Properties

Label 2-52e2-1.1-c1-0-3
Degree $2$
Conductor $2704$
Sign $1$
Analytic cond. $21.5915$
Root an. cond. $4.64667$
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  − 1.69·3-s + 1.19·5-s − 4.29·7-s − 0.137·9-s − 3.69·11-s − 2.02·15-s − 7.85·17-s + 4.65·19-s + 7.26·21-s − 3.71·23-s − 3.56·25-s + 5.30·27-s + 4.29·29-s − 2.91·31-s + 6.24·33-s − 5.14·35-s − 8.03·37-s + 8.10·41-s − 0.884·43-s − 0.164·45-s + 4.58·47-s + 11.4·49-s + 13.2·51-s + 5.43·53-s − 4.42·55-s − 7.87·57-s − 2.32·59-s + ⋯
L(s)  = 1  − 0.976·3-s + 0.535·5-s − 1.62·7-s − 0.0456·9-s − 1.11·11-s − 0.523·15-s − 1.90·17-s + 1.06·19-s + 1.58·21-s − 0.774·23-s − 0.712·25-s + 1.02·27-s + 0.797·29-s − 0.522·31-s + 1.08·33-s − 0.869·35-s − 1.32·37-s + 1.26·41-s − 0.134·43-s − 0.0244·45-s + 0.668·47-s + 1.63·49-s + 1.86·51-s + 0.746·53-s − 0.596·55-s − 1.04·57-s − 0.303·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2704\)    =    \(2^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(21.5915\)
Root analytic conductor: \(4.64667\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2704,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4647125410\)
\(L(\frac12)\) \(\approx\) \(0.4647125410\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + 1.69T + 3T^{2} \)
5 \( 1 - 1.19T + 5T^{2} \)
7 \( 1 + 4.29T + 7T^{2} \)
11 \( 1 + 3.69T + 11T^{2} \)
17 \( 1 + 7.85T + 17T^{2} \)
19 \( 1 - 4.65T + 19T^{2} \)
23 \( 1 + 3.71T + 23T^{2} \)
29 \( 1 - 4.29T + 29T^{2} \)
31 \( 1 + 2.91T + 31T^{2} \)
37 \( 1 + 8.03T + 37T^{2} \)
41 \( 1 - 8.10T + 41T^{2} \)
43 \( 1 + 0.884T + 43T^{2} \)
47 \( 1 - 4.58T + 47T^{2} \)
53 \( 1 - 5.43T + 53T^{2} \)
59 \( 1 + 2.32T + 59T^{2} \)
61 \( 1 + 6.25T + 61T^{2} \)
67 \( 1 - 3.33T + 67T^{2} \)
71 \( 1 - 4.35T + 71T^{2} \)
73 \( 1 - 3.82T + 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 + 5.24T + 83T^{2} \)
89 \( 1 - 9.46T + 89T^{2} \)
97 \( 1 - 3.67T + 97T^{2} \)
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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

−9.091338015546489537130091087006, −8.050959708839000786269604100093, −6.99776496310863435472079497796, −6.46182042634360261582873910082, −5.77182700118171215171156998211, −5.23039556361652102578464774545, −4.13373328174842081229521486464, −3.02288058373235829804022087277, −2.23108081560296737513201880713, −0.41414188101732677903154585741, 0.41414188101732677903154585741, 2.23108081560296737513201880713, 3.02288058373235829804022087277, 4.13373328174842081229521486464, 5.23039556361652102578464774545, 5.77182700118171215171156998211, 6.46182042634360261582873910082, 6.99776496310863435472079497796, 8.050959708839000786269604100093, 9.091338015546489537130091087006

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