Properties

Label 2-6004-1.1-c1-0-66
Degree $2$
Conductor $6004$
Sign $1$
Analytic cond. $47.9421$
Root an. cond. $6.92402$
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  + 1.98·3-s + 1.21·5-s + 4.02·7-s + 0.938·9-s + 2.10·11-s − 1.82·13-s + 2.40·15-s − 2.39·17-s + 19-s + 7.99·21-s + 0.824·23-s − 3.52·25-s − 4.09·27-s + 7.86·29-s + 1.02·31-s + 4.16·33-s + 4.89·35-s − 1.30·37-s − 3.62·39-s + 1.13·41-s + 12.7·43-s + 1.13·45-s − 3.49·47-s + 9.22·49-s − 4.75·51-s − 0.890·53-s + 2.55·55-s + ⋯
L(s)  = 1  + 1.14·3-s + 0.543·5-s + 1.52·7-s + 0.312·9-s + 0.633·11-s − 0.506·13-s + 0.622·15-s − 0.581·17-s + 0.229·19-s + 1.74·21-s + 0.171·23-s − 0.705·25-s − 0.787·27-s + 1.45·29-s + 0.184·31-s + 0.725·33-s + 0.826·35-s − 0.214·37-s − 0.580·39-s + 0.176·41-s + 1.95·43-s + 0.169·45-s − 0.509·47-s + 1.31·49-s − 0.666·51-s − 0.122·53-s + 0.343·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
Sign: $1$
Analytic conductor: \(47.9421\)
Root analytic conductor: \(6.92402\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.249284109\)
\(L(\frac12)\) \(\approx\) \(4.249284109\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 - T \)
79 \( 1 - T \)
good3 \( 1 - 1.98T + 3T^{2} \)
5 \( 1 - 1.21T + 5T^{2} \)
7 \( 1 - 4.02T + 7T^{2} \)
11 \( 1 - 2.10T + 11T^{2} \)
13 \( 1 + 1.82T + 13T^{2} \)
17 \( 1 + 2.39T + 17T^{2} \)
23 \( 1 - 0.824T + 23T^{2} \)
29 \( 1 - 7.86T + 29T^{2} \)
31 \( 1 - 1.02T + 31T^{2} \)
37 \( 1 + 1.30T + 37T^{2} \)
41 \( 1 - 1.13T + 41T^{2} \)
43 \( 1 - 12.7T + 43T^{2} \)
47 \( 1 + 3.49T + 47T^{2} \)
53 \( 1 + 0.890T + 53T^{2} \)
59 \( 1 - 2.48T + 59T^{2} \)
61 \( 1 - 6.87T + 61T^{2} \)
67 \( 1 - 8.07T + 67T^{2} \)
71 \( 1 - 3.03T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
83 \( 1 - 16.1T + 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + 5.70T + 97T^{2} \)
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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

−8.058246146592569397163314558132, −7.66462185431243593587001180650, −6.78466790256812134723447112271, −5.92986244993907337174200699383, −5.06944421820997693236960172733, −4.42662735319040418116332238042, −3.62792389039670244586462338649, −2.51257472099695262473936633702, −2.08826781418735670196745400685, −1.09844156461181472029050653286, 1.09844156461181472029050653286, 2.08826781418735670196745400685, 2.51257472099695262473936633702, 3.62792389039670244586462338649, 4.42662735319040418116332238042, 5.06944421820997693236960172733, 5.92986244993907337174200699383, 6.78466790256812134723447112271, 7.66462185431243593587001180650, 8.058246146592569397163314558132

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