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Properties

Label 2-1008-63.20-c1-0-28
Degree $2$
Conductor $1008$
Sign $0.227 + 0.973i$
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  + (−1.69 + 0.354i)3-s + (−0.895 − 1.55i)5-s + (2.30 − 1.30i)7-s + (2.74 − 1.20i)9-s + (2.07 + 1.20i)11-s + (−4.23 + 2.44i)13-s + (2.06 + 2.31i)15-s + 3.66·17-s − 3.01i·19-s + (−3.44 + 3.02i)21-s + (−3.26 + 1.88i)23-s + (0.897 − 1.55i)25-s + (−4.23 + 3.00i)27-s + (−5.68 − 3.28i)29-s + (4.02 − 2.32i)31-s + ⋯
L(s)  = 1  + (−0.978 + 0.204i)3-s + (−0.400 − 0.693i)5-s + (0.870 − 0.492i)7-s + (0.916 − 0.400i)9-s + (0.627 + 0.362i)11-s + (−1.17 + 0.678i)13-s + (0.533 + 0.596i)15-s + 0.888·17-s − 0.692i·19-s + (−0.750 + 0.660i)21-s + (−0.680 + 0.392i)23-s + (0.179 − 0.310i)25-s + (−0.815 + 0.579i)27-s + (−1.05 − 0.609i)29-s + (0.722 − 0.417i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.227 + 0.973i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.227 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.017908037\)
\(L(\frac12)\) \(\approx\) \(1.017908037\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 0.354i)T \)
7 \( 1 + (-2.30 + 1.30i)T \)
good5 \( 1 + (0.895 + 1.55i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.07 - 1.20i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.23 - 2.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.66T + 17T^{2} \)
19 \( 1 + 3.01iT - 19T^{2} \)
23 \( 1 + (3.26 - 1.88i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.68 + 3.28i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.02 + 2.32i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.36T + 37T^{2} \)
41 \( 1 + (4.04 + 6.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.48 + 6.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.56 - 4.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (7.29 + 12.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.81 - 5.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.285 - 0.493i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.96iT - 71T^{2} \)
73 \( 1 + 12.3iT - 73T^{2} \)
79 \( 1 + (-1.51 + 2.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.00 + 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.74T + 89T^{2} \)
97 \( 1 + (-4.77 - 2.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.780144033578311865495217824455, −9.184783850770992360253511318893, −7.88812789415650827339694144059, −7.35974320886005072732818731877, −6.34708908706186245879661301622, −5.23649593169737273733743921905, −4.56555137184873691918342917302, −3.94813137835848391179062874312, −1.92195224937477990966379571637, −0.60127032344826368128219920726, 1.28986351469069398770267297107, 2.73451217829051121512929585636, 4.04226830381315772333548462504, 5.10724361463403498454440355373, 5.79473953989169262238110157198, 6.73945499871678076385172948071, 7.65843381961727850621535880053, 8.147168857265911277184364799751, 9.559582718393500849278337024738, 10.25790444998374542756481902807

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