Properties

Label 2-1339-1.1-c1-0-76
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  + 1.93·2-s + 2.64·3-s + 1.74·4-s + 1.65·5-s + 5.11·6-s − 1.38·7-s − 0.501·8-s + 3.98·9-s + 3.20·10-s + 4.47·11-s + 4.60·12-s − 13-s − 2.67·14-s + 4.38·15-s − 4.45·16-s − 2.03·17-s + 7.70·18-s + 4.15·19-s + 2.88·20-s − 3.65·21-s + 8.65·22-s − 7.01·23-s − 1.32·24-s − 2.24·25-s − 1.93·26-s + 2.60·27-s − 2.40·28-s + ⋯
L(s)  = 1  + 1.36·2-s + 1.52·3-s + 0.870·4-s + 0.741·5-s + 2.08·6-s − 0.522·7-s − 0.177·8-s + 1.32·9-s + 1.01·10-s + 1.34·11-s + 1.32·12-s − 0.277·13-s − 0.714·14-s + 1.13·15-s − 1.11·16-s − 0.493·17-s + 1.81·18-s + 0.954·19-s + 0.645·20-s − 0.797·21-s + 1.84·22-s − 1.46·23-s − 0.270·24-s − 0.449·25-s − 0.379·26-s + 0.501·27-s − 0.454·28-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

\(L(1)\) \(\approx\) \(5.818385539\)
\(L(\frac12)\) \(\approx\) \(5.818385539\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.93T + 2T^{2} \)
3 \( 1 - 2.64T + 3T^{2} \)
5 \( 1 - 1.65T + 5T^{2} \)
7 \( 1 + 1.38T + 7T^{2} \)
11 \( 1 - 4.47T + 11T^{2} \)
17 \( 1 + 2.03T + 17T^{2} \)
19 \( 1 - 4.15T + 19T^{2} \)
23 \( 1 + 7.01T + 23T^{2} \)
29 \( 1 - 6.30T + 29T^{2} \)
31 \( 1 + 0.243T + 31T^{2} \)
37 \( 1 - 3.87T + 37T^{2} \)
41 \( 1 + 5.23T + 41T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 + 5.32T + 47T^{2} \)
53 \( 1 - 9.63T + 53T^{2} \)
59 \( 1 - 8.86T + 59T^{2} \)
61 \( 1 - 5.29T + 61T^{2} \)
67 \( 1 + 2.73T + 67T^{2} \)
71 \( 1 + 0.281T + 71T^{2} \)
73 \( 1 - 1.22T + 73T^{2} \)
79 \( 1 - 7.45T + 79T^{2} \)
83 \( 1 - 9.08T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + 0.747T + 97T^{2} \)
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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

−9.643621645937070403073819517508, −8.877184265970920373778870089747, −8.078766560013741847505694917902, −6.84076874171928346652118152098, −6.33749280124326581217317221570, −5.30651268065370440106042809221, −4.17165914550045956536452220066, −3.58696701281126169615323536435, −2.72042076167962773441319307059, −1.82071920755827205765463855465, 1.82071920755827205765463855465, 2.72042076167962773441319307059, 3.58696701281126169615323536435, 4.17165914550045956536452220066, 5.30651268065370440106042809221, 6.33749280124326581217317221570, 6.84076874171928346652118152098, 8.078766560013741847505694917902, 8.877184265970920373778870089747, 9.643621645937070403073819517508

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