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Properties

Label 2-840-105.104-c1-0-14
Degree $2$
Conductor $840$
Sign $-0.213 - 0.976i$
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  + (1.37 + 1.04i)3-s + (1.84 + 1.26i)5-s + (−2.63 − 0.269i)7-s + (0.801 + 2.89i)9-s + 3.01i·11-s − 4.39·13-s + (1.20 + 3.67i)15-s − 2.71i·17-s + 8.23i·19-s + (−3.34 − 3.13i)21-s + 3.81·23-s + (1.77 + 4.67i)25-s + (−1.92 + 4.82i)27-s − 1.17i·29-s − 1.73i·31-s + ⋯
L(s)  = 1  + (0.796 + 0.605i)3-s + (0.823 + 0.567i)5-s + (−0.994 − 0.101i)7-s + (0.267 + 0.963i)9-s + 0.908i·11-s − 1.21·13-s + (0.311 + 0.950i)15-s − 0.657i·17-s + 1.88i·19-s + (−0.730 − 0.683i)21-s + 0.796·23-s + (0.355 + 0.934i)25-s + (−0.370 + 0.928i)27-s − 0.217i·29-s − 0.311i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.213 - 0.976i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.213 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18599 + 1.47314i\)
\(L(\frac12)\) \(\approx\) \(1.18599 + 1.47314i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.37 - 1.04i)T \)
5 \( 1 + (-1.84 - 1.26i)T \)
7 \( 1 + (2.63 + 0.269i)T \)
good11 \( 1 - 3.01iT - 11T^{2} \)
13 \( 1 + 4.39T + 13T^{2} \)
17 \( 1 + 2.71iT - 17T^{2} \)
19 \( 1 - 8.23iT - 19T^{2} \)
23 \( 1 - 3.81T + 23T^{2} \)
29 \( 1 + 1.17iT - 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + 4.60iT - 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + 9.18iT - 43T^{2} \)
47 \( 1 - 12.0iT - 47T^{2} \)
53 \( 1 - 7.14T + 53T^{2} \)
59 \( 1 + 9.11T + 59T^{2} \)
61 \( 1 + 13.5iT - 61T^{2} \)
67 \( 1 + 0.494iT - 67T^{2} \)
71 \( 1 + 5.15iT - 71T^{2} \)
73 \( 1 + 4.76T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 8.16iT - 83T^{2} \)
89 \( 1 - 2.26T + 89T^{2} \)
97 \( 1 + 2.92T + 97T^{2} \)
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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

−10.18967381422532606489085934093, −9.538197499332031240733805299425, −9.252218720278030792028327249087, −7.71086881569901170449006280733, −7.20227011107980866181140383757, −6.07863752521681530303221651652, −5.07625783921144595554853709869, −3.94821817510318130962426628111, −2.89483526738255534116927098326, −2.08600194756549453750740373306, 0.825742825695920753559121112517, 2.42635070259941655298636365739, 3.09574243165483399279777073643, 4.55552703832644490698480708569, 5.71126425873743344326765421190, 6.59359781968327890169448173957, 7.28982816662994043329994663981, 8.518660494700331767212456006848, 9.092074477837289699519229654395, 9.661417799762890822294443886559

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