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Properties

Label 2-66139-1.1-c1-0-0
Degree $2$
Conductor $66139$
Sign $-1$
Analytic cond. $528.122$
Root an. cond. $22.9809$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s + 3·5-s − 7-s + 9-s − 3·11-s + 4·12-s + 4·13-s − 6·15-s + 4·16-s − 3·17-s + 19-s − 6·20-s + 2·21-s + 4·25-s + 4·27-s + 2·28-s + 6·29-s + 4·31-s + 6·33-s − 3·35-s − 2·36-s − 2·37-s − 8·39-s − 6·41-s + 43-s + 6·44-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s + 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.15·12-s + 1.10·13-s − 1.54·15-s + 16-s − 0.727·17-s + 0.229·19-s − 1.34·20-s + 0.436·21-s + 4/5·25-s + 0.769·27-s + 0.377·28-s + 1.11·29-s + 0.718·31-s + 1.04·33-s − 0.507·35-s − 1/3·36-s − 0.328·37-s − 1.28·39-s − 0.937·41-s + 0.152·43-s + 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66139 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66139 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66139\)    =    \(19 \cdot 59^{2}\)
Sign: $-1$
Analytic conductor: \(528.122\)
Root analytic conductor: \(22.9809\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 66139,\ (\ :1/2),\ -1)\)

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)$
bad19 \( 1 - T \)
59 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.10368891267580, −13.75783359973715, −13.56562716637174, −12.98129451392242, −12.56200424647394, −12.03281508644314, −11.35394520566884, −10.77096402943641, −10.41935639391799, −9.913194043072926, −9.565460304181922, −8.752850433197578, −8.503789351638388, −7.897692775726075, −6.760014330088412, −6.603185727023104, −5.933709706293072, −5.421927747694180, −5.222254453683583, −4.511501238664877, −3.871314313511923, −3.006168226492923, −2.436075677436532, −1.440644592093003, −0.8199670550314268, 0, 0.8199670550314268, 1.440644592093003, 2.436075677436532, 3.006168226492923, 3.871314313511923, 4.511501238664877, 5.222254453683583, 5.421927747694180, 5.933709706293072, 6.603185727023104, 6.760014330088412, 7.897692775726075, 8.503789351638388, 8.752850433197578, 9.565460304181922, 9.913194043072926, 10.41935639391799, 10.77096402943641, 11.35394520566884, 12.03281508644314, 12.56200424647394, 12.98129451392242, 13.56562716637174, 13.75783359973715, 14.10368891267580

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