Properties

Label 2-6042-1.1-c1-0-115
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 + 2.32·5-s + 6-s − 0.928·7-s − 8-s + 9-s − 2.32·10-s + 6.48·11-s − 12-s − 4.80·13-s + 0.928·14-s − 2.32·15-s + 16-s − 0.708·17-s − 18-s + 19-s + 2.32·20-s + 0.928·21-s − 6.48·22-s + 3.31·23-s + 24-s + 0.384·25-s + 4.80·26-s − 27-s − 0.928·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.03·5-s + 0.408·6-s − 0.351·7-s − 0.353·8-s + 0.333·9-s − 0.733·10-s + 1.95·11-s − 0.288·12-s − 1.33·13-s + 0.248·14-s − 0.599·15-s + 0.250·16-s − 0.171·17-s − 0.235·18-s + 0.229·19-s + 0.518·20-s + 0.202·21-s − 1.38·22-s + 0.690·23-s + 0.204·24-s + 0.0768·25-s + 0.941·26-s − 0.192·27-s − 0.175·28-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 \( 1 - 2.32T + 5T^{2} \)
7 \( 1 + 0.928T + 7T^{2} \)
11 \( 1 - 6.48T + 11T^{2} \)
13 \( 1 + 4.80T + 13T^{2} \)
17 \( 1 + 0.708T + 17T^{2} \)
23 \( 1 - 3.31T + 23T^{2} \)
29 \( 1 - 0.356T + 29T^{2} \)
31 \( 1 + 7.07T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + 6.64T + 41T^{2} \)
43 \( 1 + 2.77T + 43T^{2} \)
47 \( 1 - 3.23T + 47T^{2} \)
59 \( 1 + 2.56T + 59T^{2} \)
61 \( 1 - 0.142T + 61T^{2} \)
67 \( 1 + 7.24T + 67T^{2} \)
71 \( 1 - 7.50T + 71T^{2} \)
73 \( 1 - 0.749T + 73T^{2} \)
79 \( 1 - 5.37T + 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + 5.61T + 89T^{2} \)
97 \( 1 + 16.5T + 97T^{2} \)
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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

−7.54323815407831336901463848624, −6.75636040979927622054693869702, −6.64608207912720757747871348632, −5.64689928656233672258388033288, −5.08585659968923685492026391136, −4.02429777065605835562059954370, −3.09837500123150586374012742075, −1.93969769661782771733266286315, −1.38019595959850813839538452364, 0, 1.38019595959850813839538452364, 1.93969769661782771733266286315, 3.09837500123150586374012742075, 4.02429777065605835562059954370, 5.08585659968923685492026391136, 5.64689928656233672258388033288, 6.64608207912720757747871348632, 6.75636040979927622054693869702, 7.54323815407831336901463848624

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