Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 + 0.300i)3-s + (−1.26 + 2.19i)5-s + (0.5 + 0.866i)7-s + (2.81 − 1.02i)9-s + (0.233 + 0.405i)11-s + (−2.91 + 5.04i)13-s + (1.5 − 4.12i)15-s + 3.87·17-s + 2.18·19-s + (−1.11 − 1.32i)21-s + (−0.0530 + 0.0918i)23-s + (−0.705 − 1.22i)25-s + (−4.49 + 2.59i)27-s + (−4.39 − 7.60i)29-s + (−3.84 + 6.65i)31-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)3-s + (−0.566 + 0.980i)5-s + (0.188 + 0.327i)7-s + (0.939 − 0.342i)9-s + (0.0705 + 0.122i)11-s + (−0.807 + 1.39i)13-s + (0.387 − 1.06i)15-s + 0.940·17-s + 0.501·19-s + (−0.242 − 0.289i)21-s + (−0.0110 + 0.0191i)23-s + (−0.141 − 0.244i)25-s + (−0.866 + 0.499i)27-s + (−0.815 − 1.41i)29-s + (−0.689 + 1.19i)31-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (337, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(\approx\)  \(0.4969853143\)
\(L(\frac12)\)  \(\approx\)  \(0.4969853143\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.70 - 0.300i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (1.26 - 2.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.233 - 0.405i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.91 - 5.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.87T + 17T^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 + (0.0530 - 0.0918i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.39 + 7.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.84 - 6.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.68T + 37T^{2} \)
41 \( 1 + (-1.11 + 1.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.613 - 1.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.66 + 4.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.716T + 53T^{2} \)
59 \( 1 + (-0.368 + 0.637i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.479 + 0.829i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.81 - 8.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + (6.31 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.36 + 2.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.11T + 89T^{2} \)
97 \( 1 + (-6.80 - 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.41230381473354823538142171521, −9.775609260244395990654377005278, −8.858139447915015586790874563810, −7.37387828310427735713459142139, −7.20032853034883521802606103902, −6.13373739265818901579652288431, −5.22970001412823381162898142837, −4.26717112675753584662117407788, −3.28058416340649369229655902088, −1.78617183941398144140853609023, 0.27427708269657519555789863141, 1.40612509920672158778229391881, 3.29895697477240429701127352211, 4.47119294362224353242174637234, 5.25951120974827262824113446218, 5.81658589064262948397060554976, 7.30289454772301239431520128419, 7.63194189944478149141349563422, 8.630592519807894043189613004347, 9.725034417280832606976108675359

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