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Properties

Label 2-1008-21.20-c1-0-5
Degree $2$
Conductor $1008$
Sign $0.577 - 0.816i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.64·7-s + 6.57i·11-s − 1.91i·23-s − 5·25-s + 8.89i·29-s + 10.5·37-s + 5.29·43-s + 7.00·49-s + 0.412i·53-s + 4·67-s + 15.0i·71-s + 17.3i·77-s − 8·79-s − 10.4i·107-s + 10.5·109-s + ⋯
L(s)  = 1  + 0.999·7-s + 1.98i·11-s − 0.399i·23-s − 25-s + 1.65i·29-s + 1.73·37-s + 0.806·43-s + 49-s + 0.0566i·53-s + 0.488·67-s + 1.78i·71-s + 1.98i·77-s − 0.900·79-s − 1.00i·107-s + 1.01·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.685476907\)
\(L(\frac12)\) \(\approx\) \(1.685476907\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - 2.64T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 - 6.57iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 1.91iT - 23T^{2} \)
29 \( 1 - 8.89iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5.29T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 0.412iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 15.0iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.03653193485484825452889291421, −9.360038643549111066690090488967, −8.359601592821292376878951321385, −7.52651778843674376506836320186, −6.93870308876511580600497206883, −5.68239710821036298003559546593, −4.73751541789535850205978025105, −4.12010089099414563476440024272, −2.49677115775873564532924668312, −1.51829546174089979473293110301, 0.835274460908166022819766043613, 2.32813526070258951455098060767, 3.55833094069663626619975279282, 4.52081055544470500274424450765, 5.71507161321583498540315998976, 6.14823058732313091128899140780, 7.63178466754886756642576183889, 8.096953963338707993690981599879, 8.909428019792695170025692045101, 9.810536252700461952755964174633

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