Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.732 + 0.680i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (0.854 − 2.87i)3-s + 3i·4-s + (4.57 + 2.01i)5-s + (−1.42 − 2.63i)6-s + (3.94 − 5.77i)7-s + (4.94 + 4.94i)8-s + (−7.53 − 4.91i)9-s + (4.65 − 1.81i)10-s + 2.58i·11-s + (8.62 + 2.56i)12-s + (−8.94 − 8.94i)13-s + (−1.29 − 6.87i)14-s + (9.69 − 11.4i)15-s − 4.99·16-s + (−0.581 − 0.581i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.353i)2-s + (0.284 − 0.958i)3-s + 0.750i·4-s + (0.915 + 0.402i)5-s + (−0.238 − 0.439i)6-s + (0.564 − 0.825i)7-s + (0.618 + 0.618i)8-s + (−0.837 − 0.546i)9-s + (0.465 − 0.181i)10-s + 0.235i·11-s + (0.718 + 0.213i)12-s + (−0.687 − 0.687i)13-s + (−0.0924 − 0.491i)14-s + (0.646 − 0.763i)15-s − 0.312·16-s + (−0.0342 − 0.0342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.732 + 0.680i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (62, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.732 + 0.680i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.81984 - 0.714767i\)
\(L(\frac12)\)  \(\approx\)  \(1.81984 - 0.714767i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-0.854 + 2.87i)T \)
5 \( 1 + (-4.57 - 2.01i)T \)
7 \( 1 + (-3.94 + 5.77i)T \)
good2 \( 1 + (-0.707 + 0.707i)T - 4iT^{2} \)
11 \( 1 - 2.58iT - 121T^{2} \)
13 \( 1 + (8.94 + 8.94i)T + 169iT^{2} \)
17 \( 1 + (0.581 + 0.581i)T + 289iT^{2} \)
19 \( 1 + 16.5T + 361T^{2} \)
23 \( 1 + (-26.5 - 26.5i)T + 529iT^{2} \)
29 \( 1 + 11.5T + 841T^{2} \)
31 \( 1 - 30.8iT - 961T^{2} \)
37 \( 1 + (41.4 + 41.4i)T + 1.36e3iT^{2} \)
41 \( 1 - 17.1T + 1.68e3T^{2} \)
43 \( 1 + (25.1 - 25.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (45.2 + 45.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (-34.1 - 34.1i)T + 2.80e3iT^{2} \)
59 \( 1 - 47.6iT - 3.48e3T^{2} \)
61 \( 1 + 78.9iT - 3.72e3T^{2} \)
67 \( 1 + (-72.4 - 72.4i)T + 4.48e3iT^{2} \)
71 \( 1 + 49.0iT - 5.04e3T^{2} \)
73 \( 1 + (26.0 + 26.0i)T + 5.32e3iT^{2} \)
79 \( 1 + 75.8iT - 6.24e3T^{2} \)
83 \( 1 + (-53.6 + 53.6i)T - 6.88e3iT^{2} \)
89 \( 1 - 14.0iT - 7.92e3T^{2} \)
97 \( 1 + (-25.9 + 25.9i)T - 9.40e3iT^{2} \)
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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

−13.26667319862521073096525473965, −12.67429056739349281521077591677, −11.44509304219159263746868290056, −10.45532464451443061860707003466, −8.898794496487618878380725272599, −7.63615741921838713004644858033, −6.91041840669388112195376999615, −5.16904580807917990761159568726, −3.29559659433136519559037254657, −1.90301172049601354602340813326, 2.21450153540500752366099952307, 4.59607124968272394031454604051, 5.31682130163837465396707460591, 6.47801850536233869892786751067, 8.534798000246721891670130278079, 9.366949856135797804365388266927, 10.29124372274093869548015066778, 11.34558555700566488323785047194, 12.87625348630463356683438072748, 14.02236775349577819761213501488

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