Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.497 + 0.867i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.72 − 2.72i)2-s + (−1.22 + 1.22i)3-s + 10.8i·4-s + (4.39 − 2.38i)5-s + 6.67·6-s + (−1.87 − 1.87i)7-s + (18.6 − 18.6i)8-s − 2.99i·9-s + (−18.4 − 5.47i)10-s + 3.42·11-s + (−13.2 − 13.2i)12-s + (7.98 − 7.98i)13-s + 10.1i·14-s + (−2.46 + 8.30i)15-s − 57.9·16-s + (−16.5 − 16.5i)17-s + ⋯
L(s)  = 1  + (−1.36 − 1.36i)2-s + (−0.408 + 0.408i)3-s + 2.70i·4-s + (0.879 − 0.476i)5-s + 1.11·6-s + (−0.267 − 0.267i)7-s + (2.32 − 2.32i)8-s − 0.333i·9-s + (−1.84 − 0.547i)10-s + 0.311·11-s + (−1.10 − 1.10i)12-s + (0.613 − 0.613i)13-s + 0.727i·14-s + (−0.164 + 0.553i)15-s − 3.62·16-s + (−0.974 − 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.497 + 0.867i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.497 + 0.867i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.318040 - 0.548733i\)
\(L(\frac12)\)  \(\approx\)  \(0.318040 - 0.548733i\)
\(L(2)\)   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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + (-4.39 + 2.38i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good2 \( 1 + (2.72 + 2.72i)T + 4iT^{2} \)
11 \( 1 - 3.42T + 121T^{2} \)
13 \( 1 + (-7.98 + 7.98i)T - 169iT^{2} \)
17 \( 1 + (16.5 + 16.5i)T + 289iT^{2} \)
19 \( 1 - 1.38iT - 361T^{2} \)
23 \( 1 + (-18.8 + 18.8i)T - 529iT^{2} \)
29 \( 1 + 45.7iT - 841T^{2} \)
31 \( 1 - 43.2T + 961T^{2} \)
37 \( 1 + (-2.18 - 2.18i)T + 1.36e3iT^{2} \)
41 \( 1 - 6.61T + 1.68e3T^{2} \)
43 \( 1 + (44.1 - 44.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-14.2 - 14.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (44.4 - 44.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 17.2iT - 3.48e3T^{2} \)
61 \( 1 - 48.0T + 3.72e3T^{2} \)
67 \( 1 + (-40.9 - 40.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 38.7T + 5.04e3T^{2} \)
73 \( 1 + (-66.4 + 66.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 5.30iT - 6.24e3T^{2} \)
83 \( 1 + (62.5 - 62.5i)T - 6.88e3iT^{2} \)
89 \( 1 - 44.8iT - 7.92e3T^{2} \)
97 \( 1 + (30.5 + 30.5i)T + 9.40e3iT^{2} \)
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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

−12.88037659784280171481569640779, −11.76279653939642093311816233057, −10.85244529092400644277748319364, −9.952236635348907276414543199635, −9.248298634467126882672060001180, −8.242581135341014248713780083713, −6.54901639949465772751357547597, −4.45754001577770992550869358533, −2.71830777360441439920981105213, −0.841155734951819678245516687037, 1.62701005541909201789357564195, 5.32506810853360677984988483789, 6.46212938643798440050680832652, 6.87847963518197800809455539941, 8.465482854293293927066690784411, 9.290288158044828727951658679241, 10.38625211092992803448043025149, 11.26229296574130364632204002527, 13.20933280704476216766736267386, 14.15727110265631351060725195844

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