Properties

Label 2-3042-1.1-c1-0-49
Degree $2$
Conductor $3042$
Sign $-1$
Analytic cond. $24.2904$
Root an. cond. $4.92853$
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-s + 4-s + 3·5-s − 3·7-s − 8-s − 3·10-s + 3·14-s + 16-s + 3·17-s − 6·19-s + 3·20-s − 6·23-s + 4·25-s − 3·28-s − 32-s − 3·34-s − 9·35-s − 3·37-s + 6·38-s − 3·40-s + 43-s + 6·46-s + 3·47-s + 2·49-s − 4·50-s + 6·53-s + 3·56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.34·5-s − 1.13·7-s − 0.353·8-s − 0.948·10-s + 0.801·14-s + 1/4·16-s + 0.727·17-s − 1.37·19-s + 0.670·20-s − 1.25·23-s + 4/5·25-s − 0.566·28-s − 0.176·32-s − 0.514·34-s − 1.52·35-s − 0.493·37-s + 0.973·38-s − 0.474·40-s + 0.152·43-s + 0.884·46-s + 0.437·47-s + 2/7·49-s − 0.565·50-s + 0.824·53-s + 0.400·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3042\)    =    \(2 \cdot 3^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(24.2904\)
Root analytic conductor: \(4.92853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3042,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−8.571423131733288125113369380721, −7.60233865517211300096655945141, −6.75973359536125424427821286498, −6.01357873304239857580488774330, −5.77655495511337207389488133517, −4.40034260373823956119588744066, −3.27914026857724435022736802635, −2.39071591840719444382300423431, −1.54439702007463877805293824731, 0, 1.54439702007463877805293824731, 2.39071591840719444382300423431, 3.27914026857724435022736802635, 4.40034260373823956119588744066, 5.77655495511337207389488133517, 6.01357873304239857580488774330, 6.75973359536125424427821286498, 7.60233865517211300096655945141, 8.571423131733288125113369380721

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