Properties

Label 2-3042-1.1-c1-0-23
Degree $2$
Conductor $3042$
Sign $1$
Analytic cond. $24.2904$
Root an. cond. $4.92853$
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-s + 4-s + 0.267·5-s − 0.732·7-s + 8-s + 0.267·10-s + 4.73·11-s − 0.732·14-s + 16-s + 2.26·17-s − 1.26·19-s + 0.267·20-s + 4.73·22-s + 6.19·23-s − 4.92·25-s − 0.732·28-s − 2.46·29-s + 5.46·31-s + 32-s + 2.26·34-s − 0.196·35-s − 10.4·37-s − 1.26·38-s + 0.267·40-s + 11.3·41-s + 7.66·43-s + 4.73·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.119·5-s − 0.276·7-s + 0.353·8-s + 0.0847·10-s + 1.42·11-s − 0.195·14-s + 0.250·16-s + 0.550·17-s − 0.290·19-s + 0.0599·20-s + 1.00·22-s + 1.29·23-s − 0.985·25-s − 0.138·28-s − 0.457·29-s + 0.981·31-s + 0.176·32-s + 0.388·34-s − 0.0331·35-s − 1.72·37-s − 0.205·38-s + 0.0423·40-s + 1.77·41-s + 1.16·43-s + 0.713·44-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3042,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.309113222\)
\(L(\frac12)\) \(\approx\) \(3.309113222\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.267T + 5T^{2} \)
7 \( 1 + 0.732T + 7T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
17 \( 1 - 2.26T + 17T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 - 6.19T + 23T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 - 7.66T + 43T^{2} \)
47 \( 1 + 8.19T + 47T^{2} \)
53 \( 1 + 0.464T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 1.19T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 - 1.26T + 71T^{2} \)
73 \( 1 - 9.73T + 73T^{2} \)
79 \( 1 + 9.46T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + 2.53T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−8.777285779634056065292063414770, −7.84798522162871798466678426427, −6.98057238369366006138231477940, −6.42472535630975994459228731397, −5.68561993460932933947777542085, −4.81707916508546336497260291380, −3.92345352723761991098957054046, −3.31062698000435634896133731644, −2.17333292392493369761009921253, −1.06160160747160921100454593550, 1.06160160747160921100454593550, 2.17333292392493369761009921253, 3.31062698000435634896133731644, 3.92345352723761991098957054046, 4.81707916508546336497260291380, 5.68561993460932933947777542085, 6.42472535630975994459228731397, 6.98057238369366006138231477940, 7.84798522162871798466678426427, 8.777285779634056065292063414770

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