Properties

Label 2-6042-1.1-c1-0-57
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 − 4.22·5-s − 6-s + 3.89·7-s − 8-s + 9-s + 4.22·10-s + 1.39·11-s + 12-s + 6.39·13-s − 3.89·14-s − 4.22·15-s + 16-s + 6.62·17-s − 18-s + 19-s − 4.22·20-s + 3.89·21-s − 1.39·22-s + 4.91·23-s − 24-s + 12.8·25-s − 6.39·26-s + 27-s + 3.89·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.88·5-s − 0.408·6-s + 1.47·7-s − 0.353·8-s + 0.333·9-s + 1.33·10-s + 0.421·11-s + 0.288·12-s + 1.77·13-s − 1.04·14-s − 1.09·15-s + 0.250·16-s + 1.60·17-s − 0.235·18-s + 0.229·19-s − 0.944·20-s + 0.849·21-s − 0.297·22-s + 1.02·23-s − 0.204·24-s + 2.56·25-s − 1.25·26-s + 0.192·27-s + 0.735·28-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

\(L(1)\) \(\approx\) \(2.070058688\)
\(L(\frac12)\) \(\approx\) \(2.070058688\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.22T + 5T^{2} \)
7 \( 1 - 3.89T + 7T^{2} \)
11 \( 1 - 1.39T + 11T^{2} \)
13 \( 1 - 6.39T + 13T^{2} \)
17 \( 1 - 6.62T + 17T^{2} \)
23 \( 1 - 4.91T + 23T^{2} \)
29 \( 1 - 5.17T + 29T^{2} \)
31 \( 1 - 9.08T + 31T^{2} \)
37 \( 1 - 3.19T + 37T^{2} \)
41 \( 1 + 8.33T + 41T^{2} \)
43 \( 1 + 2.91T + 43T^{2} \)
47 \( 1 + 3.31T + 47T^{2} \)
59 \( 1 - 1.28T + 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 - 1.72T + 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + 8.59T + 73T^{2} \)
79 \( 1 - 0.401T + 79T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 + 3.45T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−8.230586218665400344964884061636, −7.69076037573186599243789436989, −7.03223782329680238825030599441, −6.15677895149590424130143869140, −4.94807649300013009532307761291, −4.36729743896145632693458892836, −3.45457389963519497606399714419, −3.03020171497537094475545166276, −1.35238237064917503633584467665, −1.01009749164106530204310038226, 1.01009749164106530204310038226, 1.35238237064917503633584467665, 3.03020171497537094475545166276, 3.45457389963519497606399714419, 4.36729743896145632693458892836, 4.94807649300013009532307761291, 6.15677895149590424130143869140, 7.03223782329680238825030599441, 7.69076037573186599243789436989, 8.230586218665400344964884061636

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