Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.641 - 0.766i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.67 − 0.449i)3-s + (1.87 + 3.24i)5-s + (0.5 − 0.866i)7-s + (2.59 − 1.50i)9-s + (−1.82 + 3.16i)11-s + (2.77 + 4.80i)13-s + (4.59 + 4.58i)15-s − 7.20·17-s + 3.30·19-s + (0.447 − 1.67i)21-s + (−2.49 − 4.32i)23-s + (−4.52 + 7.84i)25-s + (3.66 − 3.68i)27-s + (−0.245 + 0.425i)29-s + (−1.94 − 3.37i)31-s + ⋯
L(s)  = 1  + (0.965 − 0.259i)3-s + (0.838 + 1.45i)5-s + (0.188 − 0.327i)7-s + (0.865 − 0.501i)9-s + (−0.550 + 0.953i)11-s + (0.769 + 1.33i)13-s + (1.18 + 1.18i)15-s − 1.74·17-s + 0.758·19-s + (0.0975 − 0.365i)21-s + (−0.520 − 0.900i)23-s + (−0.905 + 1.56i)25-s + (0.705 − 0.708i)27-s + (−0.0455 + 0.0789i)29-s + (−0.349 − 0.605i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.641 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.641 - 0.766i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (673, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.641 - 0.766i)\)
\(L(1)\)  \(\approx\)  \(2.567690670\)
\(L(\frac12)\)  \(\approx\)  \(2.567690670\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.67 + 0.449i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.87 - 3.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.82 - 3.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.77 - 4.80i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.20T + 17T^{2} \)
19 \( 1 - 3.30T + 19T^{2} \)
23 \( 1 + (2.49 + 4.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.245 - 0.425i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.94 + 3.37i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.89T + 37T^{2} \)
41 \( 1 + (2.38 + 4.12i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.801 - 1.38i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.81 + 8.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.03T + 53T^{2} \)
59 \( 1 + (0.754 + 1.30i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.04 + 1.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.70 - 2.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 - 9.83T + 73T^{2} \)
79 \( 1 + (-1.86 + 3.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.69 - 9.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.14T + 89T^{2} \)
97 \( 1 + (-5.45 + 9.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.920472550160234836695447656450, −9.397681485098162105037449280341, −8.449358696669781588915460710549, −7.36440914613945504971099016907, −6.81982902854611535923023852102, −6.20832567650984727738806365176, −4.59429329526082414432558800085, −3.69460458261305999086505274716, −2.38649886124281136422978392741, −1.98054941973034237064094665683, 1.12445364140883091083139610497, 2.34195032338292114553405492504, 3.45051427316684619735547221347, 4.67355080250792694044451541595, 5.40967214841830633273977384358, 6.20868215884077431792100885708, 7.83792112315669691531598763723, 8.282267986658155939868699642387, 9.055704695599051422199274236364, 9.504969804210017019484678015096

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