Properties

Degree $2$
Conductor $6004$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.60·5-s − 2.96·7-s − 3·9-s − 6.18·11-s + 3.15·13-s − 3.39·17-s + 19-s − 4.49·23-s − 2.43·25-s − 2.60·29-s + 1.91·31-s − 4.74·35-s + 2.15·37-s + 4.04·41-s + 10.3·43-s − 4.80·45-s + 9.46·47-s + 1.76·49-s + 6.85·53-s − 9.90·55-s − 11.9·59-s − 6.13·61-s + 8.88·63-s + 5.05·65-s − 0.994·67-s + 5.98·71-s + 0.489·73-s + ⋯
L(s)  = 1  + 0.716·5-s − 1.11·7-s − 9-s − 1.86·11-s + 0.874·13-s − 0.822·17-s + 0.229·19-s − 0.936·23-s − 0.486·25-s − 0.484·29-s + 0.343·31-s − 0.801·35-s + 0.354·37-s + 0.631·41-s + 1.57·43-s − 0.716·45-s + 1.37·47-s + 0.252·49-s + 0.941·53-s − 1.33·55-s − 1.55·59-s − 0.785·61-s + 1.11·63-s + 0.626·65-s − 0.121·67-s + 0.710·71-s + 0.0573·73-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: $1$
Motivic weight: \(1\)
Character: $\chi_{6004} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.023073961\)
\(L(\frac12)\) \(\approx\) \(1.023073961\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T^{2} \)
5 \( 1 - 1.60T + 5T^{2} \)
7 \( 1 + 2.96T + 7T^{2} \)
11 \( 1 + 6.18T + 11T^{2} \)
13 \( 1 - 3.15T + 13T^{2} \)
17 \( 1 + 3.39T + 17T^{2} \)
23 \( 1 + 4.49T + 23T^{2} \)
29 \( 1 + 2.60T + 29T^{2} \)
31 \( 1 - 1.91T + 31T^{2} \)
37 \( 1 - 2.15T + 37T^{2} \)
41 \( 1 - 4.04T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 - 9.46T + 47T^{2} \)
53 \( 1 - 6.85T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 + 6.13T + 61T^{2} \)
67 \( 1 + 0.994T + 67T^{2} \)
71 \( 1 - 5.98T + 71T^{2} \)
73 \( 1 - 0.489T + 73T^{2} \)
83 \( 1 + 1.53T + 83T^{2} \)
89 \( 1 - 9.91T + 89T^{2} \)
97 \( 1 - 4.02T + 97T^{2} \)
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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

−8.000819507164175309023871668469, −7.51383975147119158133652808034, −6.39426210564919655642155127114, −5.88309773221879392881315984320, −5.55652405857742099598021479845, −4.45578797840149570874612732102, −3.48001008074339947960128434799, −2.67131906968221753793305778868, −2.16330244336412494014137388729, −0.49298851118096174713315771942, 0.49298851118096174713315771942, 2.16330244336412494014137388729, 2.67131906968221753793305778868, 3.48001008074339947960128434799, 4.45578797840149570874612732102, 5.55652405857742099598021479845, 5.88309773221879392881315984320, 6.39426210564919655642155127114, 7.51383975147119158133652808034, 8.000819507164175309023871668469

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