Properties

Label 2-1339-1.1-c1-0-3
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  − 2.15·2-s − 0.0701·3-s + 2.66·4-s − 2.25·5-s + 0.151·6-s − 1.38·7-s − 1.42·8-s − 2.99·9-s + 4.86·10-s − 4.35·11-s − 0.186·12-s − 13-s + 2.98·14-s + 0.158·15-s − 2.24·16-s + 3.87·17-s + 6.46·18-s − 7.82·19-s − 5.99·20-s + 0.0968·21-s + 9.40·22-s − 2.49·23-s + 0.100·24-s + 0.0834·25-s + 2.15·26-s + 0.420·27-s − 3.67·28-s + ⋯
L(s)  = 1  − 1.52·2-s − 0.0404·3-s + 1.33·4-s − 1.00·5-s + 0.0617·6-s − 0.521·7-s − 0.504·8-s − 0.998·9-s + 1.53·10-s − 1.31·11-s − 0.0538·12-s − 0.277·13-s + 0.796·14-s + 0.0408·15-s − 0.560·16-s + 0.939·17-s + 1.52·18-s − 1.79·19-s − 1.34·20-s + 0.0211·21-s + 2.00·22-s − 0.520·23-s + 0.0204·24-s + 0.0166·25-s + 0.423·26-s + 0.0808·27-s − 0.694·28-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

\(L(1)\) \(\approx\) \(0.1608929794\)
\(L(\frac12)\) \(\approx\) \(0.1608929794\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.15T + 2T^{2} \)
3 \( 1 + 0.0701T + 3T^{2} \)
5 \( 1 + 2.25T + 5T^{2} \)
7 \( 1 + 1.38T + 7T^{2} \)
11 \( 1 + 4.35T + 11T^{2} \)
17 \( 1 - 3.87T + 17T^{2} \)
19 \( 1 + 7.82T + 19T^{2} \)
23 \( 1 + 2.49T + 23T^{2} \)
29 \( 1 - 6.85T + 29T^{2} \)
31 \( 1 + 7.84T + 31T^{2} \)
37 \( 1 - 0.413T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 3.34T + 43T^{2} \)
47 \( 1 + 8.45T + 47T^{2} \)
53 \( 1 + 2.05T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 6.57T + 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + 1.00T + 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 - 0.796T + 79T^{2} \)
83 \( 1 - 1.36T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 - 5.58T + 97T^{2} \)
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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

−9.613919444834834909526405737037, −8.579673145249343872589895059990, −8.158530206430610793426845339795, −7.59767238990658814082395571489, −6.64182186364225607303927122468, −5.65638918067651378412742017855, −4.44818341197338425526740713568, −3.19151388887531725237173614132, −2.20666340022641749508331071524, −0.34153267758274259013425016831, 0.34153267758274259013425016831, 2.20666340022641749508331071524, 3.19151388887531725237173614132, 4.44818341197338425526740713568, 5.65638918067651378412742017855, 6.64182186364225607303927122468, 7.59767238990658814082395571489, 8.158530206430610793426845339795, 8.579673145249343872589895059990, 9.613919444834834909526405737037

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