Properties

Label 2-1004-1.1-c1-0-20
Degree $2$
Conductor $1004$
Sign $-1$
Analytic cond. $8.01698$
Root an. cond. $2.83142$
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  + 2.18·3-s − 1.66·5-s − 1.81·7-s + 1.76·9-s − 3.97·11-s − 4.88·13-s − 3.63·15-s − 3.09·17-s − 3.71·19-s − 3.96·21-s + 2.52·23-s − 2.22·25-s − 2.70·27-s + 4.16·29-s + 4.17·31-s − 8.67·33-s + 3.02·35-s + 11.0·37-s − 10.6·39-s − 0.0987·41-s − 2.04·43-s − 2.93·45-s + 0.469·47-s − 3.69·49-s − 6.75·51-s − 3.85·53-s + 6.62·55-s + ⋯
L(s)  = 1  + 1.25·3-s − 0.745·5-s − 0.686·7-s + 0.587·9-s − 1.19·11-s − 1.35·13-s − 0.939·15-s − 0.750·17-s − 0.853·19-s − 0.865·21-s + 0.525·23-s − 0.444·25-s − 0.519·27-s + 0.772·29-s + 0.750·31-s − 1.51·33-s + 0.512·35-s + 1.81·37-s − 1.70·39-s − 0.0154·41-s − 0.312·43-s − 0.437·45-s + 0.0684·47-s − 0.528·49-s − 0.946·51-s − 0.530·53-s + 0.893·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1004\)    =    \(2^{2} \cdot 251\)
Sign: $-1$
Analytic conductor: \(8.01698\)
Root analytic conductor: \(2.83142\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1004,\ (\ :1/2),\ -1)\)

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 \( 1 \)
251 \( 1 - T \)
good3 \( 1 - 2.18T + 3T^{2} \)
5 \( 1 + 1.66T + 5T^{2} \)
7 \( 1 + 1.81T + 7T^{2} \)
11 \( 1 + 3.97T + 11T^{2} \)
13 \( 1 + 4.88T + 13T^{2} \)
17 \( 1 + 3.09T + 17T^{2} \)
19 \( 1 + 3.71T + 19T^{2} \)
23 \( 1 - 2.52T + 23T^{2} \)
29 \( 1 - 4.16T + 29T^{2} \)
31 \( 1 - 4.17T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 + 0.0987T + 41T^{2} \)
43 \( 1 + 2.04T + 43T^{2} \)
47 \( 1 - 0.469T + 47T^{2} \)
53 \( 1 + 3.85T + 53T^{2} \)
59 \( 1 - 7.09T + 59T^{2} \)
61 \( 1 + 0.640T + 61T^{2} \)
67 \( 1 - 1.93T + 67T^{2} \)
71 \( 1 - 0.867T + 71T^{2} \)
73 \( 1 + 6.51T + 73T^{2} \)
79 \( 1 + 9.82T + 79T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 + 9.81T + 89T^{2} \)
97 \( 1 + 1.98T + 97T^{2} \)
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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

−9.545331865424948668765995031694, −8.584537575096230449670345623579, −7.962940473867430758259382547523, −7.33085047665568424133903620355, −6.31674425580127582622917092187, −4.93051749639337159435019982338, −4.07178723948994027199954644363, −2.88841821871650602751729776188, −2.41767108685476935599368957497, 0, 2.41767108685476935599368957497, 2.88841821871650602751729776188, 4.07178723948994027199954644363, 4.93051749639337159435019982338, 6.31674425580127582622917092187, 7.33085047665568424133903620355, 7.962940473867430758259382547523, 8.584537575096230449670345623579, 9.545331865424948668765995031694

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