Properties

Label 2-1004-1.1-c1-0-19
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  + 0.358·3-s + 2.25·5-s − 4.66·7-s − 2.87·9-s − 0.681·11-s + 0.464·13-s + 0.806·15-s − 0.986·17-s − 2.32·19-s − 1.67·21-s − 5.37·23-s + 0.0786·25-s − 2.10·27-s − 4.75·29-s − 1.57·31-s − 0.244·33-s − 10.5·35-s + 3.55·37-s + 0.166·39-s + 0.278·41-s − 1.78·43-s − 6.47·45-s + 2.22·47-s + 14.8·49-s − 0.353·51-s + 2.88·53-s − 1.53·55-s + ⋯
L(s)  = 1  + 0.206·3-s + 1.00·5-s − 1.76·7-s − 0.957·9-s − 0.205·11-s + 0.128·13-s + 0.208·15-s − 0.239·17-s − 0.534·19-s − 0.364·21-s − 1.12·23-s + 0.0157·25-s − 0.404·27-s − 0.882·29-s − 0.282·31-s − 0.0424·33-s − 1.77·35-s + 0.583·37-s + 0.0266·39-s + 0.0434·41-s − 0.271·43-s − 0.964·45-s + 0.324·47-s + 2.11·49-s − 0.0494·51-s + 0.396·53-s − 0.207·55-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 \( 1 - 0.358T + 3T^{2} \)
5 \( 1 - 2.25T + 5T^{2} \)
7 \( 1 + 4.66T + 7T^{2} \)
11 \( 1 + 0.681T + 11T^{2} \)
13 \( 1 - 0.464T + 13T^{2} \)
17 \( 1 + 0.986T + 17T^{2} \)
19 \( 1 + 2.32T + 19T^{2} \)
23 \( 1 + 5.37T + 23T^{2} \)
29 \( 1 + 4.75T + 29T^{2} \)
31 \( 1 + 1.57T + 31T^{2} \)
37 \( 1 - 3.55T + 37T^{2} \)
41 \( 1 - 0.278T + 41T^{2} \)
43 \( 1 + 1.78T + 43T^{2} \)
47 \( 1 - 2.22T + 47T^{2} \)
53 \( 1 - 2.88T + 53T^{2} \)
59 \( 1 + 6.32T + 59T^{2} \)
61 \( 1 - 0.0560T + 61T^{2} \)
67 \( 1 + 5.83T + 67T^{2} \)
71 \( 1 + 2.74T + 71T^{2} \)
73 \( 1 - 9.63T + 73T^{2} \)
79 \( 1 - 3.00T + 79T^{2} \)
83 \( 1 + 2.32T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 4.01T + 97T^{2} \)
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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

−9.522594169476215817523746586366, −8.994793320350810436753731474038, −7.968449624741711535562844629285, −6.78942175498291654781305032440, −6.03729042165850155491309914802, −5.59725008911451040959480612921, −4.00100275758084920371367759363, −3.00802294467622105437045996369, −2.13114939772750612921109865413, 0, 2.13114939772750612921109865413, 3.00802294467622105437045996369, 4.00100275758084920371367759363, 5.59725008911451040959480612921, 6.03729042165850155491309914802, 6.78942175498291654781305032440, 7.968449624741711535562844629285, 8.994793320350810436753731474038, 9.522594169476215817523746586366

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