Properties

Degree $2$
Conductor $1008$
Sign $0.5 - 0.866i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.11 + 1.32i)3-s + (0.439 − 0.761i)5-s + (0.5 + 0.866i)7-s + (−0.520 + 2.95i)9-s + (1.93 + 3.35i)11-s + (2.72 − 4.72i)13-s + (1.5 − 0.264i)15-s + 1.65·17-s − 2.41·19-s + (−0.592 + 1.62i)21-s + (1.58 − 2.73i)23-s + (2.11 + 3.66i)25-s + (−4.5 + 2.59i)27-s + (3.02 + 5.23i)29-s + (−2.27 + 3.94i)31-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)3-s + (0.196 − 0.340i)5-s + (0.188 + 0.327i)7-s + (−0.173 + 0.984i)9-s + (0.584 + 1.01i)11-s + (0.756 − 1.30i)13-s + (0.387 − 0.0682i)15-s + 0.400·17-s − 0.553·19-s + (−0.129 + 0.355i)21-s + (0.329 − 0.571i)23-s + (0.422 + 0.732i)25-s + (−0.866 + 0.499i)27-s + (0.561 + 0.972i)29-s + (−0.408 + 0.708i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.5 - 0.866i$
Motivic weight: \(1\)
Character: $\chi_{1008} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.201575050\)
\(L(\frac12)\) \(\approx\) \(2.201575050\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 - 1.32i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.439 + 0.761i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.93 - 3.35i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.72 + 4.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.65T + 17T^{2} \)
19 \( 1 + 2.41T + 19T^{2} \)
23 \( 1 + (-1.58 + 2.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.02 - 5.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.27 - 3.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.55T + 37T^{2} \)
41 \( 1 + (-0.592 + 1.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0923 - 0.160i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.511 + 0.885i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.29T + 53T^{2} \)
59 \( 1 + (-3.33 + 5.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.29 - 2.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.47 - 2.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + (2.97 + 5.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.109 + 0.189i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + (6.25 + 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.25143137163148994206572070459, −9.029739825518235027177698170882, −8.750660061213648832307443604644, −7.79635229867851623251652596321, −6.79934270521761139760556170017, −5.51106899598835049743640217072, −4.89601658221015665136827576233, −3.81649840934088093443857162008, −2.87379612677785301862097154924, −1.54739200221319459854699463644, 1.07557105603813125082445035523, 2.24280527903032127388952740519, 3.45148047845483548542918218645, 4.24306421836378867503753801782, 5.88166780282746217921209615962, 6.50419620413243778186133385311, 7.24099846412641456310032281346, 8.290703276922955241009827101882, 8.824546491935362081186182693333, 9.647581851045754290252241054099

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