Properties

Label 2-6042-1.1-c1-0-51
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 − 0.312·5-s + 6-s − 1.12·7-s + 8-s + 9-s − 0.312·10-s + 4.84·11-s + 12-s − 2.90·13-s − 1.12·14-s − 0.312·15-s + 16-s − 2.37·17-s + 18-s + 19-s − 0.312·20-s − 1.12·21-s + 4.84·22-s + 7.43·23-s + 24-s − 4.90·25-s − 2.90·26-s + 27-s − 1.12·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.139·5-s + 0.408·6-s − 0.426·7-s + 0.353·8-s + 0.333·9-s − 0.0986·10-s + 1.45·11-s + 0.288·12-s − 0.804·13-s − 0.301·14-s − 0.0805·15-s + 0.250·16-s − 0.575·17-s + 0.235·18-s + 0.229·19-s − 0.0697·20-s − 0.245·21-s + 1.03·22-s + 1.54·23-s + 0.204·24-s − 0.980·25-s − 0.568·26-s + 0.192·27-s − 0.213·28-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

\(L(1)\) \(\approx\) \(4.081867756\)
\(L(\frac12)\) \(\approx\) \(4.081867756\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.312T + 5T^{2} \)
7 \( 1 + 1.12T + 7T^{2} \)
11 \( 1 - 4.84T + 11T^{2} \)
13 \( 1 + 2.90T + 13T^{2} \)
17 \( 1 + 2.37T + 17T^{2} \)
23 \( 1 - 7.43T + 23T^{2} \)
29 \( 1 - 2.54T + 29T^{2} \)
31 \( 1 + 1.67T + 31T^{2} \)
37 \( 1 + 4.39T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 - 6.43T + 43T^{2} \)
47 \( 1 - 9.40T + 47T^{2} \)
59 \( 1 - 1.03T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 + 3.92T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + 3.56T + 73T^{2} \)
79 \( 1 + 2.83T + 79T^{2} \)
83 \( 1 - 9.84T + 83T^{2} \)
89 \( 1 + 4.79T + 89T^{2} \)
97 \( 1 - 4.44T + 97T^{2} \)
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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

−7.931055122389247441576136292713, −7.16562111160319652441835655997, −6.76821635479800098691063557109, −5.96906013545899853877353816855, −5.09436934496963354694225284838, −4.24835512209240473840365178436, −3.75034053545048603217901198087, −2.87651861608429791113740918739, −2.11679605850612478447534576307, −0.953089253865494263116087313569, 0.953089253865494263116087313569, 2.11679605850612478447534576307, 2.87651861608429791113740918739, 3.75034053545048603217901198087, 4.24835512209240473840365178436, 5.09436934496963354694225284838, 5.96906013545899853877353816855, 6.76821635479800098691063557109, 7.16562111160319652441835655997, 7.931055122389247441576136292713

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