Properties

Label 2-1339-1.1-c1-0-0
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  − 1.52·2-s − 3.31·3-s + 0.310·4-s + 0.643·5-s + 5.03·6-s − 3.98·7-s + 2.56·8-s + 7.97·9-s − 0.977·10-s − 5.02·11-s − 1.02·12-s − 13-s + 6.05·14-s − 2.13·15-s − 4.52·16-s − 1.32·17-s − 12.1·18-s + 5.39·19-s + 0.199·20-s + 13.2·21-s + 7.63·22-s − 1.98·23-s − 8.50·24-s − 4.58·25-s + 1.52·26-s − 16.4·27-s − 1.23·28-s + ⋯
L(s)  = 1  − 1.07·2-s − 1.91·3-s + 0.155·4-s + 0.287·5-s + 2.05·6-s − 1.50·7-s + 0.907·8-s + 2.65·9-s − 0.309·10-s − 1.51·11-s − 0.297·12-s − 0.277·13-s + 1.61·14-s − 0.550·15-s − 1.13·16-s − 0.321·17-s − 2.85·18-s + 1.23·19-s + 0.0446·20-s + 2.88·21-s + 1.62·22-s − 0.413·23-s − 1.73·24-s − 0.917·25-s + 0.298·26-s − 3.17·27-s − 0.234·28-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

\(L(1)\) \(\approx\) \(0.04957283884\)
\(L(\frac12)\) \(\approx\) \(0.04957283884\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.52T + 2T^{2} \)
3 \( 1 + 3.31T + 3T^{2} \)
5 \( 1 - 0.643T + 5T^{2} \)
7 \( 1 + 3.98T + 7T^{2} \)
11 \( 1 + 5.02T + 11T^{2} \)
17 \( 1 + 1.32T + 17T^{2} \)
19 \( 1 - 5.39T + 19T^{2} \)
23 \( 1 + 1.98T + 23T^{2} \)
29 \( 1 + 6.50T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + 4.06T + 37T^{2} \)
41 \( 1 - 0.282T + 41T^{2} \)
43 \( 1 - 1.38T + 43T^{2} \)
47 \( 1 + 13.1T + 47T^{2} \)
53 \( 1 + 2.72T + 53T^{2} \)
59 \( 1 + 0.751T + 59T^{2} \)
61 \( 1 + 9.39T + 61T^{2} \)
67 \( 1 + 9.59T + 67T^{2} \)
71 \( 1 - 0.812T + 71T^{2} \)
73 \( 1 + 2.34T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 - 6.39T + 83T^{2} \)
89 \( 1 - 4.25T + 89T^{2} \)
97 \( 1 + 3.54T + 97T^{2} \)
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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

−9.682901549879020116556739249931, −9.318280403700304609849538842301, −7.68237886715300452216266725446, −7.29581152767015211407726615505, −6.30108223184745084532683143323, −5.54347466907034433612419158694, −4.91175304291185106376551652796, −3.58588237268669021886686434889, −1.81408588502400900516078337492, −0.20319390225061309421418684888, 0.20319390225061309421418684888, 1.81408588502400900516078337492, 3.58588237268669021886686434889, 4.91175304291185106376551652796, 5.54347466907034433612419158694, 6.30108223184745084532683143323, 7.29581152767015211407726615505, 7.68237886715300452216266725446, 9.318280403700304609849538842301, 9.682901549879020116556739249931

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