Properties

Label 2-1339-1.1-c1-0-37
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  + 2.05·2-s − 2.28·3-s + 2.21·4-s + 0.331·5-s − 4.68·6-s + 3.06·7-s + 0.444·8-s + 2.20·9-s + 0.679·10-s + 1.18·11-s − 5.05·12-s + 13-s + 6.30·14-s − 0.755·15-s − 3.52·16-s + 3.46·17-s + 4.52·18-s − 1.47·19-s + 0.733·20-s − 6.99·21-s + 2.43·22-s + 4.02·23-s − 1.01·24-s − 4.89·25-s + 2.05·26-s + 1.81·27-s + 6.80·28-s + ⋯
L(s)  = 1  + 1.45·2-s − 1.31·3-s + 1.10·4-s + 0.148·5-s − 1.91·6-s + 1.15·7-s + 0.157·8-s + 0.734·9-s + 0.214·10-s + 0.357·11-s − 1.45·12-s + 0.277·13-s + 1.68·14-s − 0.194·15-s − 0.880·16-s + 0.840·17-s + 1.06·18-s − 0.338·19-s + 0.164·20-s − 1.52·21-s + 0.519·22-s + 0.838·23-s − 0.206·24-s − 0.978·25-s + 0.402·26-s + 0.349·27-s + 1.28·28-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

\(L(1)\) \(\approx\) \(2.735290382\)
\(L(\frac12)\) \(\approx\) \(2.735290382\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.05T + 2T^{2} \)
3 \( 1 + 2.28T + 3T^{2} \)
5 \( 1 - 0.331T + 5T^{2} \)
7 \( 1 - 3.06T + 7T^{2} \)
11 \( 1 - 1.18T + 11T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 1.47T + 19T^{2} \)
23 \( 1 - 4.02T + 23T^{2} \)
29 \( 1 - 5.56T + 29T^{2} \)
31 \( 1 - 5.32T + 31T^{2} \)
37 \( 1 - 7.42T + 37T^{2} \)
41 \( 1 + 0.520T + 41T^{2} \)
43 \( 1 - 2.91T + 43T^{2} \)
47 \( 1 - 2.51T + 47T^{2} \)
53 \( 1 - 4.86T + 53T^{2} \)
59 \( 1 + 2.03T + 59T^{2} \)
61 \( 1 - 3.54T + 61T^{2} \)
67 \( 1 - 4.82T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + 0.954T + 79T^{2} \)
83 \( 1 - 3.48T + 83T^{2} \)
89 \( 1 + 1.29T + 89T^{2} \)
97 \( 1 + 0.434T + 97T^{2} \)
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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

−9.919975523821179763680256535743, −8.729210803776518710473146039472, −7.75037499042932397119920596645, −6.64117431994894675418180621599, −6.03809323080150615562899606169, −5.31281527113416974298945318094, −4.72403836609189739959587329264, −3.95308405699535851572683503472, −2.61578578013696519768690682648, −1.12495505966587508110403174908, 1.12495505966587508110403174908, 2.61578578013696519768690682648, 3.95308405699535851572683503472, 4.72403836609189739959587329264, 5.31281527113416974298945318094, 6.03809323080150615562899606169, 6.64117431994894675418180621599, 7.75037499042932397119920596645, 8.729210803776518710473146039472, 9.919975523821179763680256535743

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