Properties

Degree 2
Conductor $ 2^{2} \cdot 7 $
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  + (−0.5 − 0.866i)3-s + (−1.5 + 2.59i)5-s + (−2 − 1.73i)7-s + (1 − 1.73i)9-s + (1.5 + 2.59i)11-s + 2·13-s + 3·15-s + (−1.5 − 2.59i)17-s + (0.5 − 0.866i)19-s + (−0.499 + 2.59i)21-s + (−1.5 + 2.59i)23-s + (−2 − 3.46i)25-s − 5·27-s − 6·29-s + (3.5 + 6.06i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.670 + 1.16i)5-s + (−0.755 − 0.654i)7-s + (0.333 − 0.577i)9-s + (0.452 + 0.783i)11-s + 0.554·13-s + 0.774·15-s + (−0.363 − 0.630i)17-s + (0.114 − 0.198i)19-s + (−0.109 + 0.566i)21-s + (−0.312 + 0.541i)23-s + (−0.400 − 0.692i)25-s − 0.962·27-s − 1.11·29-s + (0.628 + 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(28\)    =    \(2^{2} \cdot 7\)
\( \varepsilon \)  =  $0.991 + 0.126i$
motivic weight  =  \(1\)
character  :  $\chi_{28} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 28,\ (\ :1/2),\ 0.991 + 0.126i)$
$L(1)$  $\approx$  $0.597311 - 0.0379054i$
$L(\frac12)$  $\approx$  $0.597311 - 0.0379054i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 + (2 + 1.73i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−17.52539665179351686539607658141, −15.95920936655884041627259306018, −14.93563706892529383383121625487, −13.54814894465822888147983726342, −12.20439698718194328052721890579, −11.02460749959139304874986304333, −9.616510756305673496977057334962, −7.32269192359830248395197349962, −6.59765781472494882013944777804, −3.68820621838161426787695488127, 4.16290105624966053923484697414, 5.88053399239330996514645357185, 8.202084321555248126345415607740, 9.378231197034825499779622084931, 11.04513563684512874615408261657, 12.33996226575045532080751057895, 13.40005263692909520970142355282, 15.34293187880635901845287645069, 16.22593751048373275065867264340, 16.83240693466222331245255824153

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