Properties

Label 2-1339-1.1-c1-0-38
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  − 2.35·2-s + 1.07·3-s + 3.52·4-s + 3.29·5-s − 2.52·6-s + 3.84·7-s − 3.59·8-s − 1.84·9-s − 7.75·10-s − 3.17·11-s + 3.79·12-s + 13-s − 9.05·14-s + 3.54·15-s + 1.39·16-s + 0.572·17-s + 4.34·18-s − 6.09·19-s + 11.6·20-s + 4.13·21-s + 7.46·22-s + 8.94·23-s − 3.86·24-s + 5.87·25-s − 2.35·26-s − 5.20·27-s + 13.5·28-s + ⋯
L(s)  = 1  − 1.66·2-s + 0.620·3-s + 1.76·4-s + 1.47·5-s − 1.03·6-s + 1.45·7-s − 1.27·8-s − 0.615·9-s − 2.45·10-s − 0.957·11-s + 1.09·12-s + 0.277·13-s − 2.41·14-s + 0.914·15-s + 0.348·16-s + 0.138·17-s + 1.02·18-s − 1.39·19-s + 2.60·20-s + 0.902·21-s + 1.59·22-s + 1.86·23-s − 0.788·24-s + 1.17·25-s − 0.461·26-s − 1.00·27-s + 2.56·28-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

\(L(1)\) \(\approx\) \(1.351650937\)
\(L(\frac12)\) \(\approx\) \(1.351650937\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.35T + 2T^{2} \)
3 \( 1 - 1.07T + 3T^{2} \)
5 \( 1 - 3.29T + 5T^{2} \)
7 \( 1 - 3.84T + 7T^{2} \)
11 \( 1 + 3.17T + 11T^{2} \)
17 \( 1 - 0.572T + 17T^{2} \)
19 \( 1 + 6.09T + 19T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 - 7.43T + 29T^{2} \)
31 \( 1 + 2.51T + 31T^{2} \)
37 \( 1 - 6.12T + 37T^{2} \)
41 \( 1 - 6.84T + 41T^{2} \)
43 \( 1 + 3.57T + 43T^{2} \)
47 \( 1 - 8.72T + 47T^{2} \)
53 \( 1 - 6.83T + 53T^{2} \)
59 \( 1 + 9.41T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 - 1.60T + 71T^{2} \)
73 \( 1 - 7.15T + 73T^{2} \)
79 \( 1 + 4.65T + 79T^{2} \)
83 \( 1 - 2.45T + 83T^{2} \)
89 \( 1 - 0.362T + 89T^{2} \)
97 \( 1 + 2.74T + 97T^{2} \)
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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

−9.322782502508407299325461210944, −8.862648305824626746817822205914, −8.241320823394223001911391720735, −7.63331393231378715379137824481, −6.55698512010057340246263322081, −5.62949094721821203265361168032, −4.70767768593096509024727392910, −2.67475381792004393680435647518, −2.19591348363137141299267099607, −1.12752477597494398207527936655, 1.12752477597494398207527936655, 2.19591348363137141299267099607, 2.67475381792004393680435647518, 4.70767768593096509024727392910, 5.62949094721821203265361168032, 6.55698512010057340246263322081, 7.63331393231378715379137824481, 8.241320823394223001911391720735, 8.862648305824626746817822205914, 9.322782502508407299325461210944

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