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Properties

Label 2-1008-21.20-c3-0-41
Degree $2$
Conductor $1008$
Sign $-0.577 + 0.816i$
Analytic cond. $59.4739$
Root an. cond. $7.71193$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 18.5·7-s − 66.7i·11-s − 125. i·23-s − 125·25-s + 69.7i·29-s + 10.5·37-s − 534.·43-s + 343.·49-s + 65.4i·53-s − 740·67-s − 205. i·71-s − 1.23e3i·77-s + 1.38e3·79-s − 2.21e3i·107-s − 2.27e3·109-s + ⋯
L(s)  = 1  + 0.999·7-s − 1.83i·11-s − 1.13i·23-s − 25-s + 0.446i·29-s + 0.0470·37-s − 1.89·43-s + 1.00·49-s + 0.169i·53-s − 1.34·67-s − 0.343i·71-s − 1.83i·77-s + 1.97·79-s − 1.99i·107-s − 1.99·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.477462732\)
\(L(\frac12)\) \(\approx\) \(1.477462732\)
\(L(\frac{5}{2})\) 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 \( 1 - 18.5T \)
good5 \( 1 + 125T^{2} \)
11 \( 1 + 66.7iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 + 125. iT - 1.21e4T^{2} \)
29 \( 1 - 69.7iT - 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 - 10.5T + 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 + 534.T + 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 65.4iT - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 + 740T + 3.00e5T^{2} \)
71 \( 1 + 205. iT - 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 - 1.38e3T + 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−9.105622633914310481724068837501, −8.383598023077135785122393074072, −7.88454495920508508585169510410, −6.67049825011642766385262182499, −5.81446750068822199024486566055, −4.98996654800365685366797100322, −3.91514838040862257614726789518, −2.87893353161918047997806388759, −1.57944814553310491519739368754, −0.35664526162582606058131626621, 1.47959799100561819451823429127, 2.23583348000587864502845044046, 3.77217767063025502969140843994, 4.68848026202805631642537356747, 5.37784956836820422288787778817, 6.59389063155157192653256609005, 7.52895534831360522612454788901, 7.991697406091549046222092681284, 9.156963047694362255076249004583, 9.859088449568794877755497228443

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