Properties

Label 2-1339-1.1-c1-0-40
Degree $2$
Conductor $1339$
Sign $1$
Analytic cond. $10.6919$
Root an. cond. $3.26985$
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  − 0.736·2-s + 2.43·3-s − 1.45·4-s + 4.11·5-s − 1.79·6-s − 5.11·7-s + 2.54·8-s + 2.93·9-s − 3.03·10-s + 0.686·11-s − 3.54·12-s − 13-s + 3.77·14-s + 10.0·15-s + 1.03·16-s + 5.03·17-s − 2.16·18-s − 0.596·19-s − 5.99·20-s − 12.4·21-s − 0.506·22-s + 0.406·23-s + 6.20·24-s + 11.9·25-s + 0.736·26-s − 0.163·27-s + 7.46·28-s + ⋯
L(s)  = 1  − 0.520·2-s + 1.40·3-s − 0.728·4-s + 1.84·5-s − 0.732·6-s − 1.93·7-s + 0.900·8-s + 0.977·9-s − 0.958·10-s + 0.207·11-s − 1.02·12-s − 0.277·13-s + 1.00·14-s + 2.58·15-s + 0.259·16-s + 1.22·17-s − 0.509·18-s − 0.136·19-s − 1.34·20-s − 2.72·21-s − 0.107·22-s + 0.0847·23-s + 1.26·24-s + 2.38·25-s + 0.144·26-s − 0.0315·27-s + 1.40·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $1$
Analytic conductor: \(10.6919\)
Root analytic conductor: \(3.26985\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1339,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.078613216\)
\(L(\frac12)\) \(\approx\) \(2.078613216\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + T \)
103 \( 1 - T \)
good2 \( 1 + 0.736T + 2T^{2} \)
3 \( 1 - 2.43T + 3T^{2} \)
5 \( 1 - 4.11T + 5T^{2} \)
7 \( 1 + 5.11T + 7T^{2} \)
11 \( 1 - 0.686T + 11T^{2} \)
17 \( 1 - 5.03T + 17T^{2} \)
19 \( 1 + 0.596T + 19T^{2} \)
23 \( 1 - 0.406T + 23T^{2} \)
29 \( 1 - 5.31T + 29T^{2} \)
31 \( 1 - 7.14T + 31T^{2} \)
37 \( 1 - 6.45T + 37T^{2} \)
41 \( 1 - 4.79T + 41T^{2} \)
43 \( 1 - 4.23T + 43T^{2} \)
47 \( 1 + 1.96T + 47T^{2} \)
53 \( 1 - 7.46T + 53T^{2} \)
59 \( 1 + 0.896T + 59T^{2} \)
61 \( 1 + 15.0T + 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 - 4.60T + 71T^{2} \)
73 \( 1 + 0.196T + 73T^{2} \)
79 \( 1 - 3.77T + 79T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 - 1.02T + 97T^{2} \)
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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

−9.383614027294864620089838883282, −9.226011082740208514464215352614, −8.319854255359429408094167351333, −7.31698408448086747511081370025, −6.32044097843227100301989896578, −5.65671285317805329401276191308, −4.31130711252766563842929951521, −3.10384470702915441754290660604, −2.58831578256114911711996427719, −1.15078353642078456867590379271, 1.15078353642078456867590379271, 2.58831578256114911711996427719, 3.10384470702915441754290660604, 4.31130711252766563842929951521, 5.65671285317805329401276191308, 6.32044097843227100301989896578, 7.31698408448086747511081370025, 8.319854255359429408094167351333, 9.226011082740208514464215352614, 9.383614027294864620089838883282

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