Properties

Degree 2
Conductor $ 3 \cdot 7 $
Sign $0.880 + 0.473i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.124 − 0.214i)2-s + (1.5 − 2.59i)3-s + (3.96 − 6.87i)4-s + (6.21 + 10.7i)5-s − 0.744·6-s + (−18.4 + 1.73i)7-s − 3.95·8-s + (−4.5 − 7.79i)9-s + (1.54 − 2.67i)10-s + (−30.1 + 52.2i)11-s + (−11.9 − 20.6i)12-s + 36.4·13-s + (2.66 + 3.74i)14-s + 37.3·15-s + (−31.2 − 54.1i)16-s + (24.3 − 42.2i)17-s + ⋯
L(s)  = 1  + (−0.0438 − 0.0759i)2-s + (0.288 − 0.499i)3-s + (0.496 − 0.859i)4-s + (0.556 + 0.963i)5-s − 0.0506·6-s + (−0.995 + 0.0938i)7-s − 0.174·8-s + (−0.166 − 0.288i)9-s + (0.0487 − 0.0844i)10-s + (−0.826 + 1.43i)11-s + (−0.286 − 0.496i)12-s + 0.777·13-s + (0.0507 + 0.0715i)14-s + 0.642·15-s + (−0.488 − 0.846i)16-s + (0.347 − 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $0.880 + 0.473i$
motivic weight  =  \(3\)
character  :  $\chi_{21} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 21,\ (\ :3/2),\ 0.880 + 0.473i)$
$L(2)$  $\approx$  $1.18131 - 0.297209i$
$L(\frac12)$  $\approx$  $1.18131 - 0.297209i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 + (18.4 - 1.73i)T \)
good2 \( 1 + (0.124 + 0.214i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-6.21 - 10.7i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (30.1 - 52.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 36.4T + 2.19e3T^{2} \)
17 \( 1 + (-24.3 + 42.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-25.2 - 43.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (69.3 + 120. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 61.1T + 2.43e4T^{2} \)
31 \( 1 + (-0.584 + 1.01i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (34.7 + 60.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 308.T + 6.89e4T^{2} \)
43 \( 1 - 174.T + 7.95e4T^{2} \)
47 \( 1 + (194. + 337. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (157. - 272. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (422. - 731. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-169. - 293. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-485. + 841. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 98.4T + 3.57e5T^{2} \)
73 \( 1 + (355. - 615. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-243. - 421. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 605.T + 5.71e5T^{2} \)
89 \( 1 + (109. + 188. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 782.T + 9.12e5T^{2} \)
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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

−18.18892700008772531938131117749, −16.09129734937073203102080876666, −14.94093638176937574859696846384, −13.85550663964881382335246218730, −12.40634587593307672436147169363, −10.57294761895014104508982770498, −9.670961843150247179522507042355, −7.18989616090266510259301698319, −6.03416664316828508044427140253, −2.48372776722578918437887004989, 3.38228381260687071477557465693, 5.88817185904019278055773599757, 8.119892828999079687461125020766, 9.332721178444415412381209580040, 11.05263346182170649797912524248, 12.82073607332888322503371272846, 13.58223739559200347317134805619, 15.89661175917865023681722252124, 16.23055622637122962884479404939, 17.42840234217564586993226542844

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