Properties

Label 2-6028-1.1-c1-0-93
Degree $2$
Conductor $6028$
Sign $-1$
Analytic cond. $48.1338$
Root an. cond. $6.93785$
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  + 2.59·3-s − 2.40·5-s + 0.934·7-s + 3.73·9-s + 11-s − 3.65·13-s − 6.24·15-s − 0.402·17-s − 2.79·19-s + 2.42·21-s − 3.78·23-s + 0.798·25-s + 1.89·27-s − 1.66·29-s + 9.72·31-s + 2.59·33-s − 2.25·35-s − 3.96·37-s − 9.48·39-s − 4.88·41-s + 0.416·43-s − 8.98·45-s + 7.36·47-s − 6.12·49-s − 1.04·51-s − 5.18·53-s − 2.40·55-s + ⋯
L(s)  = 1  + 1.49·3-s − 1.07·5-s + 0.353·7-s + 1.24·9-s + 0.301·11-s − 1.01·13-s − 1.61·15-s − 0.0975·17-s − 0.641·19-s + 0.529·21-s − 0.790·23-s + 0.159·25-s + 0.365·27-s − 0.309·29-s + 1.74·31-s + 0.451·33-s − 0.380·35-s − 0.651·37-s − 1.51·39-s − 0.763·41-s + 0.0634·43-s − 1.33·45-s + 1.07·47-s − 0.875·49-s − 0.146·51-s − 0.711·53-s − 0.324·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6028\)    =    \(2^{2} \cdot 11 \cdot 137\)
Sign: $-1$
Analytic conductor: \(48.1338\)
Root analytic conductor: \(6.93785\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6028,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 - T \)
137 \( 1 + T \)
good3 \( 1 - 2.59T + 3T^{2} \)
5 \( 1 + 2.40T + 5T^{2} \)
7 \( 1 - 0.934T + 7T^{2} \)
13 \( 1 + 3.65T + 13T^{2} \)
17 \( 1 + 0.402T + 17T^{2} \)
19 \( 1 + 2.79T + 19T^{2} \)
23 \( 1 + 3.78T + 23T^{2} \)
29 \( 1 + 1.66T + 29T^{2} \)
31 \( 1 - 9.72T + 31T^{2} \)
37 \( 1 + 3.96T + 37T^{2} \)
41 \( 1 + 4.88T + 41T^{2} \)
43 \( 1 - 0.416T + 43T^{2} \)
47 \( 1 - 7.36T + 47T^{2} \)
53 \( 1 + 5.18T + 53T^{2} \)
59 \( 1 - 1.14T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 9.39T + 67T^{2} \)
71 \( 1 + 5.79T + 71T^{2} \)
73 \( 1 - 2.34T + 73T^{2} \)
79 \( 1 + 0.664T + 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−7.979135666140348655287838344598, −7.24691167592234184190471966013, −6.62803805447773164660160837465, −5.45452308941806980675998932621, −4.35200835002734064808111890948, −4.12691088963562440304927673889, −3.15564871879322347369459473675, −2.49331099578865169713908263051, −1.58495084410383424361514845478, 0, 1.58495084410383424361514845478, 2.49331099578865169713908263051, 3.15564871879322347369459473675, 4.12691088963562440304927673889, 4.35200835002734064808111890948, 5.45452308941806980675998932621, 6.62803805447773164660160837465, 7.24691167592234184190471966013, 7.979135666140348655287838344598

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