Properties

Label 2-6042-1.1-c1-0-1
Degree $2$
Conductor $6042$
Sign $1$
Analytic cond. $48.2456$
Root an. cond. $6.94590$
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 − 3-s + 4-s − 0.186·5-s + 6-s − 2.28·7-s − 8-s + 9-s + 0.186·10-s − 2.54·11-s − 12-s − 3.08·13-s + 2.28·14-s + 0.186·15-s + 16-s + 0.350·17-s − 18-s − 19-s − 0.186·20-s + 2.28·21-s + 2.54·22-s + 3.42·23-s + 24-s − 4.96·25-s + 3.08·26-s − 27-s − 2.28·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0835·5-s + 0.408·6-s − 0.862·7-s − 0.353·8-s + 0.333·9-s + 0.0590·10-s − 0.767·11-s − 0.288·12-s − 0.854·13-s + 0.609·14-s + 0.0482·15-s + 0.250·16-s + 0.0849·17-s − 0.235·18-s − 0.229·19-s − 0.0417·20-s + 0.497·21-s + 0.542·22-s + 0.714·23-s + 0.204·24-s − 0.993·25-s + 0.604·26-s − 0.192·27-s − 0.431·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6042\)    =    \(2 \cdot 3 \cdot 19 \cdot 53\)
Sign: $1$
Analytic conductor: \(48.2456\)
Root analytic conductor: \(6.94590\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6042,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3584181443\)
\(L(\frac12)\) \(\approx\) \(0.3584181443\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
19 \( 1 + T \)
53 \( 1 - T \)
good5 \( 1 + 0.186T + 5T^{2} \)
7 \( 1 + 2.28T + 7T^{2} \)
11 \( 1 + 2.54T + 11T^{2} \)
13 \( 1 + 3.08T + 13T^{2} \)
17 \( 1 - 0.350T + 17T^{2} \)
23 \( 1 - 3.42T + 23T^{2} \)
29 \( 1 + 3.84T + 29T^{2} \)
31 \( 1 - 0.257T + 31T^{2} \)
37 \( 1 + 0.640T + 37T^{2} \)
41 \( 1 + 5.74T + 41T^{2} \)
43 \( 1 - 6.19T + 43T^{2} \)
47 \( 1 + 6.20T + 47T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 3.02T + 61T^{2} \)
67 \( 1 + 0.411T + 67T^{2} \)
71 \( 1 - 4.85T + 71T^{2} \)
73 \( 1 + 7.21T + 73T^{2} \)
79 \( 1 + 1.68T + 79T^{2} \)
83 \( 1 + 7.08T + 83T^{2} \)
89 \( 1 - 1.55T + 89T^{2} \)
97 \( 1 + 5.00T + 97T^{2} \)
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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

−7.944602682182720359582135842202, −7.40337828078126631752113921689, −6.76140190964852068746015329971, −6.03396480970899319339140973023, −5.35949348532548447812477627259, −4.55112501102123652759910108908, −3.47984632966418002176211248439, −2.69601434949686808590367355545, −1.72312463618480209611960700353, −0.34699169338110983737723536800, 0.34699169338110983737723536800, 1.72312463618480209611960700353, 2.69601434949686808590367355545, 3.47984632966418002176211248439, 4.55112501102123652759910108908, 5.35949348532548447812477627259, 6.03396480970899319339140973023, 6.76140190964852068746015329971, 7.40337828078126631752113921689, 7.944602682182720359582135842202

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