Properties

Label 2-6042-1.1-c1-0-126
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 0.744·5-s − 6-s − 0.530·7-s + 8-s + 9-s + 0.744·10-s − 1.93·11-s − 12-s − 3.52·13-s − 0.530·14-s − 0.744·15-s + 16-s − 0.585·17-s + 18-s + 19-s + 0.744·20-s + 0.530·21-s − 1.93·22-s + 6.08·23-s − 24-s − 4.44·25-s − 3.52·26-s − 27-s − 0.530·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.332·5-s − 0.408·6-s − 0.200·7-s + 0.353·8-s + 0.333·9-s + 0.235·10-s − 0.582·11-s − 0.288·12-s − 0.978·13-s − 0.141·14-s − 0.192·15-s + 0.250·16-s − 0.141·17-s + 0.235·18-s + 0.229·19-s + 0.166·20-s + 0.115·21-s − 0.411·22-s + 1.26·23-s − 0.204·24-s − 0.889·25-s − 0.691·26-s − 0.192·27-s − 0.100·28-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: \(1\)
Selberg data: \((2,\ 6042,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
19 \( 1 - T \)
53 \( 1 + T \)
good5 \( 1 - 0.744T + 5T^{2} \)
7 \( 1 + 0.530T + 7T^{2} \)
11 \( 1 + 1.93T + 11T^{2} \)
13 \( 1 + 3.52T + 13T^{2} \)
17 \( 1 + 0.585T + 17T^{2} \)
23 \( 1 - 6.08T + 23T^{2} \)
29 \( 1 - 8.24T + 29T^{2} \)
31 \( 1 + 10.6T + 31T^{2} \)
37 \( 1 + 3.64T + 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + 3.08T + 43T^{2} \)
47 \( 1 + 0.785T + 47T^{2} \)
59 \( 1 + 8.73T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 - 7.13T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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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

−7.46351122976386371119479026681, −6.93501383593353251318126643017, −6.18598300944663404186163810446, −5.44352225600754241151264118008, −4.97549552736776235476469860929, −4.23323479340560202450401893405, −3.18785707470660497751574789458, −2.48137724712774803097346858956, −1.44309448470157258780314808659, 0, 1.44309448470157258780314808659, 2.48137724712774803097346858956, 3.18785707470660497751574789458, 4.23323479340560202450401893405, 4.97549552736776235476469860929, 5.44352225600754241151264118008, 6.18598300944663404186163810446, 6.93501383593353251318126643017, 7.46351122976386371119479026681

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