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Properties

Label 2-840-56.27-c1-0-53
Degree $2$
Conductor $840$
Sign $-0.895 + 0.444i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.704i)2-s − i·3-s + (1.00 + 1.72i)4-s + 5-s + (−0.704 + 1.22i)6-s + (2.37 − 1.16i)7-s + (−0.0147 − 2.82i)8-s − 9-s + (−1.22 − 0.704i)10-s − 2.29·11-s + (1.72 − 1.00i)12-s − 6.22·13-s + (−3.73 − 0.247i)14-s − i·15-s + (−1.97 + 3.47i)16-s − 2.99i·17-s + ⋯
L(s)  = 1  + (−0.866 − 0.498i)2-s − 0.577i·3-s + (0.503 + 0.864i)4-s + 0.447·5-s + (−0.287 + 0.500i)6-s + (0.898 − 0.439i)7-s + (−0.00520 − 0.999i)8-s − 0.333·9-s + (−0.387 − 0.222i)10-s − 0.692·11-s + (0.498 − 0.290i)12-s − 1.72·13-s + (−0.997 − 0.0662i)14-s − 0.258i·15-s + (−0.493 + 0.869i)16-s − 0.725i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180623 - 0.770069i\)
\(L(\frac12)\) \(\approx\) \(0.180623 - 0.770069i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.704i)T \)
3 \( 1 + iT \)
5 \( 1 - T \)
7 \( 1 + (-2.37 + 1.16i)T \)
good11 \( 1 + 2.29T + 11T^{2} \)
13 \( 1 + 6.22T + 13T^{2} \)
17 \( 1 + 2.99iT - 17T^{2} \)
19 \( 1 - 0.901iT - 19T^{2} \)
23 \( 1 + 8.15iT - 23T^{2} \)
29 \( 1 + 1.15iT - 29T^{2} \)
31 \( 1 + 1.82T + 31T^{2} \)
37 \( 1 + 1.80iT - 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + 9.48T + 47T^{2} \)
53 \( 1 + 4.44iT - 53T^{2} \)
59 \( 1 - 5.71iT - 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 - 6.42T + 67T^{2} \)
71 \( 1 + 7.96iT - 71T^{2} \)
73 \( 1 - 4.10iT - 73T^{2} \)
79 \( 1 - 14.9iT - 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + 1.36iT - 89T^{2} \)
97 \( 1 + 8.12iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.925802265203422048598228378341, −9.022170541319643931970585732931, −8.092725259566945418874552950899, −7.44412856536555551801443141080, −6.80243473309690460417635323286, −5.39360603864912891436869102708, −4.38631995064271537387089649732, −2.69769137098185960528163323457, −2.04813421056610577585799843548, −0.48581909924306731004743807406, 1.72990098847475118329851531203, 2.80611035837287958047696333781, 4.73304845418907089546419934129, 5.28873956748407686797983172203, 6.16640004477817022957935157899, 7.50639305689490629221911778573, 7.921763157684776883286360157363, 9.030943435021351412965069719735, 9.622552563947609966009862901313, 10.32687135910707858297144100994

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