Properties

Label 2-2008-1.1-c1-0-31
Degree $2$
Conductor $2008$
Sign $1$
Analytic cond. $16.0339$
Root an. cond. $4.00424$
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  + 3.32·3-s − 0.316·5-s + 0.268·7-s + 8.06·9-s − 0.366·11-s − 3.69·13-s − 1.05·15-s + 5.91·17-s + 7.61·19-s + 0.891·21-s − 6.61·23-s − 4.89·25-s + 16.8·27-s + 5.48·29-s − 4.15·31-s − 1.21·33-s − 0.0849·35-s + 8.30·37-s − 12.2·39-s + 3.92·41-s + 6.01·43-s − 2.55·45-s + 1.79·47-s − 6.92·49-s + 19.6·51-s − 10.5·53-s + 0.116·55-s + ⋯
L(s)  = 1  + 1.92·3-s − 0.141·5-s + 0.101·7-s + 2.68·9-s − 0.110·11-s − 1.02·13-s − 0.272·15-s + 1.43·17-s + 1.74·19-s + 0.194·21-s − 1.37·23-s − 0.979·25-s + 3.23·27-s + 1.01·29-s − 0.746·31-s − 0.212·33-s − 0.0143·35-s + 1.36·37-s − 1.96·39-s + 0.612·41-s + 0.917·43-s − 0.380·45-s + 0.261·47-s − 0.989·49-s + 2.75·51-s − 1.44·53-s + 0.0156·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $1$
Analytic conductor: \(16.0339\)
Root analytic conductor: \(4.00424\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.665842185\)
\(L(\frac12)\) \(\approx\) \(3.665842185\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.32T + 3T^{2} \)
5 \( 1 + 0.316T + 5T^{2} \)
7 \( 1 - 0.268T + 7T^{2} \)
11 \( 1 + 0.366T + 11T^{2} \)
13 \( 1 + 3.69T + 13T^{2} \)
17 \( 1 - 5.91T + 17T^{2} \)
19 \( 1 - 7.61T + 19T^{2} \)
23 \( 1 + 6.61T + 23T^{2} \)
29 \( 1 - 5.48T + 29T^{2} \)
31 \( 1 + 4.15T + 31T^{2} \)
37 \( 1 - 8.30T + 37T^{2} \)
41 \( 1 - 3.92T + 41T^{2} \)
43 \( 1 - 6.01T + 43T^{2} \)
47 \( 1 - 1.79T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + 3.99T + 59T^{2} \)
61 \( 1 - 1.88T + 61T^{2} \)
67 \( 1 - 1.28T + 67T^{2} \)
71 \( 1 - 8.15T + 71T^{2} \)
73 \( 1 + 16.0T + 73T^{2} \)
79 \( 1 - 7.93T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + 1.39T + 89T^{2} \)
97 \( 1 + 9.53T + 97T^{2} \)
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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

−9.252465736435746111530940750170, −8.186948498096147369198246899269, −7.62407377745408878993700930212, −7.44023493087879233022511075188, −6.01809508540594575933771232128, −4.88776293037290358249689752517, −3.96282833796213972904952339130, −3.17572144949442882213082487349, −2.45002900466335728307924191255, −1.33097009806752734996327133804, 1.33097009806752734996327133804, 2.45002900466335728307924191255, 3.17572144949442882213082487349, 3.96282833796213972904952339130, 4.88776293037290358249689752517, 6.01809508540594575933771232128, 7.44023493087879233022511075188, 7.62407377745408878993700930212, 8.186948498096147369198246899269, 9.252465736435746111530940750170

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