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Properties

Label 2-1008-21.20-c1-0-10
Degree $2$
Conductor $1008$
Sign $0.716 + 0.698i$
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  + 1.53·5-s + (0.414 − 2.61i)7-s − 0.585i·11-s + 2.16i·13-s + 5.86·17-s − 5.22i·19-s + 2.24i·23-s − 2.65·25-s − 5.41i·29-s − 4.32i·31-s + (0.634 − 4i)35-s + 4·37-s + 8.92·41-s − 10.4·43-s + 7.39·47-s + ⋯
L(s)  = 1  + 0.684·5-s + (0.156 − 0.987i)7-s − 0.176i·11-s + 0.600i·13-s + 1.42·17-s − 1.19i·19-s + 0.467i·23-s − 0.531·25-s − 1.00i·29-s − 0.777i·31-s + (0.107 − 0.676i)35-s + 0.657·37-s + 1.39·41-s − 1.59·43-s + 1.07·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.716 + 0.698i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.716 + 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.849320869\)
\(L(\frac12)\) \(\approx\) \(1.849320869\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.414 + 2.61i)T \)
good5 \( 1 - 1.53T + 5T^{2} \)
11 \( 1 + 0.585iT - 11T^{2} \)
13 \( 1 - 2.16iT - 13T^{2} \)
17 \( 1 - 5.86T + 17T^{2} \)
19 \( 1 + 5.22iT - 19T^{2} \)
23 \( 1 - 2.24iT - 23T^{2} \)
29 \( 1 + 5.41iT - 29T^{2} \)
31 \( 1 + 4.32iT - 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 8.92T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 - 7.39T + 47T^{2} \)
53 \( 1 - 5.41iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 9.65T + 67T^{2} \)
71 \( 1 - 4.58iT - 71T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 - 5.86T + 89T^{2} \)
97 \( 1 + 8.28iT - 97T^{2} \)
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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

−9.732823644615020455147102759473, −9.338053058729107558744817523501, −8.034388470771734565791941853429, −7.42205675973586739405604242610, −6.43610956678431681940129061410, −5.61023679040729220766239367263, −4.55858252784311560724558724901, −3.61347517446041846520424856331, −2.30371925162194826674678194064, −0.950517270968322911644530638124, 1.45918243763939807438092904076, 2.61507622180717270589630644302, 3.67878395041024523805485258077, 5.17854964115624286983590436641, 5.64492539769875528808664121786, 6.50567586669958736008308275172, 7.74739716749173901900777506191, 8.359910925662010180838426424666, 9.358410007113588816493611612292, 9.995284318137745449485199200834

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