Properties

Label 2-6028-1.1-c1-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.04·3-s − 2.78·5-s − 0.254·7-s − 1.90·9-s + 11-s − 6.22·13-s − 2.90·15-s − 3.13·17-s + 7.51·19-s − 0.265·21-s + 3.12·23-s + 2.76·25-s − 5.12·27-s + 4.02·29-s − 6.29·31-s + 1.04·33-s + 0.708·35-s − 4.98·37-s − 6.49·39-s + 3.79·41-s + 11.5·43-s + 5.32·45-s − 9.46·47-s − 6.93·49-s − 3.27·51-s + 0.938·53-s − 2.78·55-s + ⋯
L(s)  = 1  + 0.602·3-s − 1.24·5-s − 0.0960·7-s − 0.636·9-s + 0.301·11-s − 1.72·13-s − 0.751·15-s − 0.761·17-s + 1.72·19-s − 0.0579·21-s + 0.652·23-s + 0.553·25-s − 0.986·27-s + 0.748·29-s − 1.13·31-s + 0.181·33-s + 0.119·35-s − 0.819·37-s − 1.04·39-s + 0.592·41-s + 1.76·43-s + 0.793·45-s − 1.38·47-s − 0.990·49-s − 0.459·51-s + 0.128·53-s − 0.375·55-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: \(0\)
Selberg data: \((2,\ 6028,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.115365021\)
\(L(\frac12)\) \(\approx\) \(1.115365021\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.04T + 3T^{2} \)
5 \( 1 + 2.78T + 5T^{2} \)
7 \( 1 + 0.254T + 7T^{2} \)
13 \( 1 + 6.22T + 13T^{2} \)
17 \( 1 + 3.13T + 17T^{2} \)
19 \( 1 - 7.51T + 19T^{2} \)
23 \( 1 - 3.12T + 23T^{2} \)
29 \( 1 - 4.02T + 29T^{2} \)
31 \( 1 + 6.29T + 31T^{2} \)
37 \( 1 + 4.98T + 37T^{2} \)
41 \( 1 - 3.79T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + 9.46T + 47T^{2} \)
53 \( 1 - 0.938T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 8.76T + 61T^{2} \)
67 \( 1 + 7.75T + 67T^{2} \)
71 \( 1 + 4.39T + 71T^{2} \)
73 \( 1 + 5.47T + 73T^{2} \)
79 \( 1 + 4.67T + 79T^{2} \)
83 \( 1 - 3.77T + 83T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 - 3.52T + 97T^{2} \)
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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

−7.88056896181773000089957969545, −7.49914894593707699491885929766, −7.01076801010096805067314547098, −5.92007052603163491756279588244, −4.99785993577369396959804863839, −4.46735653169169527171465040714, −3.39503875039927793003868168490, −3.03148967775497101384506509596, −2.02657265114896969521026135262, −0.51238411822065010486526362211, 0.51238411822065010486526362211, 2.02657265114896969521026135262, 3.03148967775497101384506509596, 3.39503875039927793003868168490, 4.46735653169169527171465040714, 4.99785993577369396959804863839, 5.92007052603163491756279588244, 7.01076801010096805067314547098, 7.49914894593707699491885929766, 7.88056896181773000089957969545

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