Properties

Label 2-52e2-1.1-c1-0-13
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  + 0.554·3-s − 1.44·5-s − 2.04·7-s − 2.69·9-s + 2.55·11-s − 0.801·15-s − 5.29·17-s + 5.85·19-s − 1.13·21-s + 1.89·23-s − 2.91·25-s − 3.15·27-s + 2.26·29-s + 4.26·31-s + 1.41·33-s + 2.96·35-s + 5.35·37-s + 1.27·41-s − 6.13·43-s + 3.89·45-s + 2.95·47-s − 2.80·49-s − 2.93·51-s + 5.52·53-s − 3.69·55-s + 3.24·57-s + 12.2·59-s + ⋯
L(s)  = 1  + 0.320·3-s − 0.646·5-s − 0.774·7-s − 0.897·9-s + 0.770·11-s − 0.207·15-s − 1.28·17-s + 1.34·19-s − 0.248·21-s + 0.394·23-s − 0.582·25-s − 0.607·27-s + 0.421·29-s + 0.766·31-s + 0.246·33-s + 0.500·35-s + 0.880·37-s + 0.198·41-s − 0.935·43-s + 0.579·45-s + 0.430·47-s − 0.400·49-s − 0.411·51-s + 0.758·53-s − 0.497·55-s + 0.430·57-s + 1.58·59-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

\(L(1)\) \(\approx\) \(1.348957140\)
\(L(\frac12)\) \(\approx\) \(1.348957140\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.554T + 3T^{2} \)
5 \( 1 + 1.44T + 5T^{2} \)
7 \( 1 + 2.04T + 7T^{2} \)
11 \( 1 - 2.55T + 11T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 - 5.85T + 19T^{2} \)
23 \( 1 - 1.89T + 23T^{2} \)
29 \( 1 - 2.26T + 29T^{2} \)
31 \( 1 - 4.26T + 31T^{2} \)
37 \( 1 - 5.35T + 37T^{2} \)
41 \( 1 - 1.27T + 41T^{2} \)
43 \( 1 + 6.13T + 43T^{2} \)
47 \( 1 - 2.95T + 47T^{2} \)
53 \( 1 - 5.52T + 53T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 - 8.56T + 61T^{2} \)
67 \( 1 + 0.576T + 67T^{2} \)
71 \( 1 - 4.59T + 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 + 7.72T + 83T^{2} \)
89 \( 1 - 6.61T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−8.803345263277505065107876594777, −8.200975353817394364422090609004, −7.30365333729900024525244393335, −6.59816169899930260865276582861, −5.87229261537774000294497409759, −4.84446196645026678137401890645, −3.87732708951100156659993680530, −3.21906139507664646762548936717, −2.31017403843322865421265639347, −0.69396060965967324262008402908, 0.69396060965967324262008402908, 2.31017403843322865421265639347, 3.21906139507664646762548936717, 3.87732708951100156659993680530, 4.84446196645026678137401890645, 5.87229261537774000294497409759, 6.59816169899930260865276582861, 7.30365333729900024525244393335, 8.200975353817394364422090609004, 8.803345263277505065107876594777

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