Properties

Degree 2
Conductor $ 3 \cdot 7 $
Sign $-0.907 + 0.420i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.65 − 4.59i)2-s + (1.5 − 2.59i)3-s + (−10.0 + 17.4i)4-s + (−2.78 − 4.81i)5-s − 15.9·6-s + (9.67 − 15.7i)7-s + 64.6·8-s + (−4.5 − 7.79i)9-s + (−14.7 + 25.5i)10-s + (6.95 − 12.0i)11-s + (30.2 + 52.4i)12-s + 38.6·13-s + (−98.2 − 2.58i)14-s − 16.6·15-s + (−90.8 − 157. i)16-s + (−21.7 + 37.6i)17-s + ⋯
L(s)  = 1  + (−0.938 − 1.62i)2-s + (0.288 − 0.499i)3-s + (−1.26 + 2.18i)4-s + (−0.248 − 0.430i)5-s − 1.08·6-s + (0.522 − 0.852i)7-s + 2.85·8-s + (−0.166 − 0.288i)9-s + (−0.466 + 0.808i)10-s + (0.190 − 0.330i)11-s + (0.728 + 1.26i)12-s + 0.825·13-s + (−1.87 − 0.0492i)14-s − 0.287·15-s + (−1.41 − 2.45i)16-s + (−0.310 + 0.537i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $-0.907 + 0.420i$
motivic weight  =  \(3\)
character  :  $\chi_{21} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 21,\ (\ :3/2),\ -0.907 + 0.420i)$
$L(2)$  $\approx$  $0.155592 - 0.705778i$
$L(\frac12)$  $\approx$  $0.155592 - 0.705778i$
$L(\frac{5}{2})$   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 \( 1 + (-9.67 + 15.7i)T \)
good2 \( 1 + (2.65 + 4.59i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (2.78 + 4.81i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-6.95 + 12.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 38.6T + 2.19e3T^{2} \)
17 \( 1 + (21.7 - 37.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-54.5 - 94.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-37.4 - 64.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 72.3T + 2.43e4T^{2} \)
31 \( 1 + (32.0 - 55.4i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (94.3 + 163. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 24.7T + 6.89e4T^{2} \)
43 \( 1 + 243.T + 7.95e4T^{2} \)
47 \( 1 + (-310. - 537. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-143. + 249. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (262. - 454. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-191. - 332. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (99.0 - 171. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 785.T + 3.57e5T^{2} \)
73 \( 1 + (-165. + 286. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (218. + 379. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 241.T + 5.71e5T^{2} \)
89 \( 1 + (792. + 1.37e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 79.2T + 9.12e5T^{2} \)
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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

−17.63272420443613148193145671645, −16.50760683403070963855342646629, −13.95774330046048302303212951498, −12.82956343157525426518215724326, −11.60999065433045810935077944484, −10.48528527387765232793952545781, −8.868632507400261035981468343767, −7.79543120621363543950968364786, −3.77821328450074966821513454856, −1.28503207381873994848961598039, 5.16970437169280649720345940778, 6.93249795294349278054850478660, 8.445349222325137440194941910230, 9.418284138730015966557182287993, 11.09123214643329951154746411851, 13.81182816056850186796853355307, 15.10418676117232267397733354385, 15.52455015159332867040198216850, 16.84840095460031742415751686528, 18.12302561324404978337198702412

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