Properties

Label 2-1339-1.1-c1-0-96
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.33·2-s + 2.00·3-s − 0.215·4-s − 3.88·5-s + 2.67·6-s + 2.54·7-s − 2.95·8-s + 1.00·9-s − 5.18·10-s − 3.83·11-s − 0.431·12-s − 13-s + 3.40·14-s − 7.77·15-s − 3.52·16-s − 3.24·17-s + 1.34·18-s + 1.25·19-s + 0.836·20-s + 5.09·21-s − 5.12·22-s − 3.59·23-s − 5.92·24-s + 10.0·25-s − 1.33·26-s − 3.99·27-s − 0.548·28-s + ⋯
L(s)  = 1  + 0.944·2-s + 1.15·3-s − 0.107·4-s − 1.73·5-s + 1.09·6-s + 0.962·7-s − 1.04·8-s + 0.334·9-s − 1.64·10-s − 1.15·11-s − 0.124·12-s − 0.277·13-s + 0.909·14-s − 2.00·15-s − 0.880·16-s − 0.786·17-s + 0.316·18-s + 0.287·19-s + 0.187·20-s + 1.11·21-s − 1.09·22-s − 0.749·23-s − 1.20·24-s + 2.01·25-s − 0.261·26-s − 0.768·27-s − 0.103·28-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: \(1\)
Selberg data: \((2,\ 1339,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.33T + 2T^{2} \)
3 \( 1 - 2.00T + 3T^{2} \)
5 \( 1 + 3.88T + 5T^{2} \)
7 \( 1 - 2.54T + 7T^{2} \)
11 \( 1 + 3.83T + 11T^{2} \)
17 \( 1 + 3.24T + 17T^{2} \)
19 \( 1 - 1.25T + 19T^{2} \)
23 \( 1 + 3.59T + 23T^{2} \)
29 \( 1 + 4.68T + 29T^{2} \)
31 \( 1 - 5.40T + 31T^{2} \)
37 \( 1 + 4.58T + 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 + 4.94T + 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 + 4.12T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + 8.63T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 7.80T + 73T^{2} \)
79 \( 1 + 2.73T + 79T^{2} \)
83 \( 1 - 0.267T + 83T^{2} \)
89 \( 1 - 5.09T + 89T^{2} \)
97 \( 1 + 1.27T + 97T^{2} \)
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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

−8.753328539952437637656489348638, −8.369201013610742682784239857575, −7.79980477922392927309124274178, −6.99300009798001035940047920172, −5.45876773834669287535308200436, −4.70802955026612021923964867333, −3.97576947454792228763201593628, −3.26199974570145708613186132330, −2.31700665059939183616560458405, 0, 2.31700665059939183616560458405, 3.26199974570145708613186132330, 3.97576947454792228763201593628, 4.70802955026612021923964867333, 5.45876773834669287535308200436, 6.99300009798001035940047920172, 7.79980477922392927309124274178, 8.369201013610742682784239857575, 8.753328539952437637656489348638

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