Properties

Degree $2$
Conductor $1008$
Sign $0.605 + 0.795i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)5-s + (−2.5 − 0.866i)7-s + (3 − 5.19i)11-s − 3·13-s + (2 − 3.46i)17-s + (−2.5 − 4.33i)19-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s + 4·29-s + (3.5 − 6.06i)31-s + (−1.00 − 5.19i)35-s + (4.5 + 7.79i)37-s + 2·41-s + 43-s + (−1 − 1.73i)47-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (−0.944 − 0.327i)7-s + (0.904 − 1.56i)11-s − 0.832·13-s + (0.485 − 0.840i)17-s + (−0.573 − 0.993i)19-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s + 0.742·29-s + (0.628 − 1.08i)31-s + (−0.169 − 0.878i)35-s + (0.739 + 1.28i)37-s + 0.312·41-s + 0.152·43-s + (−0.145 − 0.252i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.605 + 0.795i$
Motivic weight: \(1\)
Character: $\chi_{1008} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.443034865\)
\(L(\frac12)\) \(\approx\) \(1.443034865\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4 + 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.5 - 12.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (4 + 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.769858902041848524187846930721, −9.247718076365400481914575016681, −8.205687515322217319418128667470, −7.05789296144251081477554203990, −6.51333271219226954514676867315, −5.77973276086698090919173261824, −4.49683100950254419439239739680, −3.24351860671157406808041880531, −2.68472768168333848800309426797, −0.70667779511792272747013891799, 1.41258495170933294239625370695, 2.57874523151774041534991501304, 3.99649392325846079452411594199, 4.79284101767179958778806531719, 5.86642245507459681783434218931, 6.63492407878902111299673062076, 7.50765665145583404753682417190, 8.657473803990557409709855779032, 9.322889715259796338036299168142, 9.947016432548702552357678611376

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