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Properties

Label 2-1008-21.17-c1-0-14
Degree $2$
Conductor $1008$
Sign $-0.985 + 0.170i$
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  + (−2.09 − 3.62i)5-s + (1.62 + 2.09i)7-s + (−2.59 − 1.5i)11-s − 2.44i·13-s + (0.507 − 0.878i)17-s + (0.878 − 0.507i)19-s + (−3.67 + 2.12i)23-s + (−6.24 + 10.8i)25-s + 1.24i·29-s + (−4.86 − 2.80i)31-s + (4.18 − 10.2i)35-s + (−4.12 − 7.13i)37-s − 2.02·41-s − 8.24·43-s + (−0.507 − 0.878i)47-s + ⋯
L(s)  = 1  + (−0.935 − 1.61i)5-s + (0.612 + 0.790i)7-s + (−0.783 − 0.452i)11-s − 0.679i·13-s + (0.123 − 0.213i)17-s + (0.201 − 0.116i)19-s + (−0.766 + 0.442i)23-s + (−1.24 + 2.16i)25-s + 0.230i·29-s + (−0.873 − 0.504i)31-s + (0.706 − 1.73i)35-s + (−0.677 − 1.17i)37-s − 0.316·41-s − 1.25·43-s + (−0.0739 − 0.128i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.985 + 0.170i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.985 + 0.170i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5632498252\)
\(L(\frac12)\) \(\approx\) \(0.5632498252\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 2.09i)T \)
good5 \( 1 + (2.09 + 3.62i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + (-0.507 + 0.878i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.878 + 0.507i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.67 - 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.24iT - 29T^{2} \)
31 \( 1 + (4.86 + 2.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.12 + 7.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.02T + 41T^{2} \)
43 \( 1 + 8.24T + 43T^{2} \)
47 \( 1 + (0.507 + 0.878i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.07 + 0.621i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.76 - 9.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.12 + 2.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (-7.24 - 4.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.62 + 9.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.16T + 83T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.76iT - 97T^{2} \)
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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

−9.288624553168152738631653036677, −8.651081490874670812176775777028, −8.030478351893978193461852557252, −7.44500189569206387900591432560, −5.66269444438001996715294128248, −5.31193889005566446062549028129, −4.38528903136118230080731041973, −3.27106416151633387646515901078, −1.73227408597882765855469300667, −0.24945057345493626010700310107, 1.98141155146285295867765879712, 3.22572426215788596880903661471, 4.02395772353382131557993697405, 4.99607007008983682255944433258, 6.44946177245144953092221382121, 7.04871677445837802094528929617, 7.76841988800410628725247746342, 8.362367115752138059702785120659, 9.911315619258738903784019047898, 10.39420546316545221619666339529

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