Properties

Degree $2$
Conductor $1008$
Sign $0.991 - 0.126i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 0.866i)5-s + 2.64·7-s + (3.96 + 2.29i)11-s + 3.46i·13-s + (−4.5 − 2.59i)17-s + (1.32 + 2.29i)19-s + (3.96 − 2.29i)23-s + (−1 + 1.73i)25-s + (−1.32 + 2.29i)31-s + (3.96 − 2.29i)35-s + (−3.5 − 6.06i)37-s + 3.46i·41-s − 9.16i·43-s + (3.96 + 6.87i)47-s + 7.00·49-s + ⋯
L(s)  = 1  + (0.670 − 0.387i)5-s + 0.999·7-s + (1.19 + 0.690i)11-s + 0.960i·13-s + (−1.09 − 0.630i)17-s + (0.303 + 0.525i)19-s + (0.827 − 0.477i)23-s + (−0.200 + 0.346i)25-s + (−0.237 + 0.411i)31-s + (0.670 − 0.387i)35-s + (−0.575 − 0.996i)37-s + 0.541i·41-s − 1.39i·43-s + (0.578 + 1.00i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.991 - 0.126i$
Motivic weight: \(1\)
Character: $\chi_{1008} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.115665273\)
\(L(\frac12)\) \(\approx\) \(2.115665273\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.96 - 2.29i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (4.5 + 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.32 - 2.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.96 + 2.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (1.32 - 2.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 + 9.16iT - 43T^{2} \)
47 \( 1 + (-3.96 - 6.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.96 + 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.96 - 2.29i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.16iT - 71T^{2} \)
73 \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.96 - 2.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.46iT - 97T^{2} \)
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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

−9.783168150100075503798626046598, −9.100523809772954772429508176414, −8.615683952140882336754887959745, −7.27556897760004803016544550522, −6.74439535691993880593196688754, −5.56603854341519971068724878277, −4.71846521011306307977979935820, −3.95333522225174300825059342518, −2.20974371425420272304315663410, −1.40949386715488600927214932143, 1.20590323516215075444832225952, 2.42433849975433527769903920059, 3.64106911829651362794749256939, 4.74605152955831451290131254109, 5.72377639599791445657553349901, 6.47662414254567900730104175892, 7.40219822223365340816047759884, 8.459174659670076817924752917333, 8.989646640230076948410052316194, 10.00966489573113557064137425659

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