Properties

Label 2-5239-1.1-c1-0-330
Degree $2$
Conductor $5239$
Sign $-1$
Analytic cond. $41.8336$
Root an. cond. $6.46789$
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  − 1.48·2-s + 2.99·3-s + 0.218·4-s + 0.692·5-s − 4.46·6-s − 1.15·7-s + 2.65·8-s + 5.98·9-s − 1.03·10-s − 2.69·11-s + 0.655·12-s + 1.71·14-s + 2.07·15-s − 4.38·16-s + 4.04·17-s − 8.91·18-s − 5.06·19-s + 0.151·20-s − 3.45·21-s + 4.01·22-s + 1.01·23-s + 7.95·24-s − 4.52·25-s + 8.94·27-s − 0.251·28-s − 8.56·29-s − 3.09·30-s + ⋯
L(s)  = 1  − 1.05·2-s + 1.73·3-s + 0.109·4-s + 0.309·5-s − 1.82·6-s − 0.435·7-s + 0.938·8-s + 1.99·9-s − 0.326·10-s − 0.812·11-s + 0.189·12-s + 0.458·14-s + 0.536·15-s − 1.09·16-s + 0.980·17-s − 2.10·18-s − 1.16·19-s + 0.0338·20-s − 0.753·21-s + 0.855·22-s + 0.212·23-s + 1.62·24-s − 0.904·25-s + 1.72·27-s − 0.0476·28-s − 1.59·29-s − 0.564·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5239\)    =    \(13^{2} \cdot 31\)
Sign: $-1$
Analytic conductor: \(41.8336\)
Root analytic conductor: \(6.46789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5239,\ (\ :1/2),\ -1)\)

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 \( 1 \)
31 \( 1 - T \)
good2 \( 1 + 1.48T + 2T^{2} \)
3 \( 1 - 2.99T + 3T^{2} \)
5 \( 1 - 0.692T + 5T^{2} \)
7 \( 1 + 1.15T + 7T^{2} \)
11 \( 1 + 2.69T + 11T^{2} \)
17 \( 1 - 4.04T + 17T^{2} \)
19 \( 1 + 5.06T + 19T^{2} \)
23 \( 1 - 1.01T + 23T^{2} \)
29 \( 1 + 8.56T + 29T^{2} \)
37 \( 1 + 5.88T + 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 3.63T + 43T^{2} \)
47 \( 1 - 0.848T + 47T^{2} \)
53 \( 1 - 1.67T + 53T^{2} \)
59 \( 1 + 1.68T + 59T^{2} \)
61 \( 1 - 2.00T + 61T^{2} \)
67 \( 1 + 8.32T + 67T^{2} \)
71 \( 1 + 5.15T + 71T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 - 0.873T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 - 3.90T + 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−7.942553676142320980328860063584, −7.58141307520595233758614327116, −6.85474921053335068545030419923, −5.73955536256514967746465060114, −4.74828992929971634598922509180, −3.82144534924980896820374411075, −3.17108387623622104839942682571, −2.15327530666983735484896126536, −1.59529824775446997348684160442, 0, 1.59529824775446997348684160442, 2.15327530666983735484896126536, 3.17108387623622104839942682571, 3.82144534924980896820374411075, 4.74828992929971634598922509180, 5.73955536256514967746465060114, 6.85474921053335068545030419923, 7.58141307520595233758614327116, 7.942553676142320980328860063584

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