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Properties

Label 2-840-24.11-c1-0-18
Degree $2$
Conductor $840$
Sign $-0.999 + 0.0412i$
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  + (0.252 + 1.39i)2-s + (1.52 + 0.825i)3-s + (−1.87 + 0.701i)4-s − 5-s + (−0.764 + 2.32i)6-s + i·7-s + (−1.44 − 2.42i)8-s + (1.63 + 2.51i)9-s + (−0.252 − 1.39i)10-s + 3.18i·11-s + (−3.43 − 0.476i)12-s + 0.611i·13-s + (−1.39 + 0.252i)14-s + (−1.52 − 0.825i)15-s + (3.01 − 2.62i)16-s + 3.24i·17-s + ⋯
L(s)  = 1  + (0.178 + 0.983i)2-s + (0.879 + 0.476i)3-s + (−0.936 + 0.350i)4-s − 0.447·5-s + (−0.312 + 0.950i)6-s + 0.377i·7-s + (−0.512 − 0.858i)8-s + (0.545 + 0.837i)9-s + (−0.0797 − 0.440i)10-s + 0.959i·11-s + (−0.990 − 0.137i)12-s + 0.169i·13-s + (−0.371 + 0.0674i)14-s + (−0.393 − 0.213i)15-s + (0.753 − 0.657i)16-s + 0.786i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.999 + 0.0412i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.999 + 0.0412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0324711 - 1.57242i\)
\(L(\frac12)\) \(\approx\) \(0.0324711 - 1.57242i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.252 - 1.39i)T \)
3 \( 1 + (-1.52 - 0.825i)T \)
5 \( 1 + T \)
7 \( 1 - iT \)
good11 \( 1 - 3.18iT - 11T^{2} \)
13 \( 1 - 0.611iT - 13T^{2} \)
17 \( 1 - 3.24iT - 17T^{2} \)
19 \( 1 + 1.51T + 19T^{2} \)
23 \( 1 + 3.27T + 23T^{2} \)
29 \( 1 + 6.32T + 29T^{2} \)
31 \( 1 + 5.58iT - 31T^{2} \)
37 \( 1 + 1.51iT - 37T^{2} \)
41 \( 1 - 4.12iT - 41T^{2} \)
43 \( 1 + 0.874T + 43T^{2} \)
47 \( 1 - 0.289T + 47T^{2} \)
53 \( 1 - 3.74T + 53T^{2} \)
59 \( 1 + 4.18iT - 59T^{2} \)
61 \( 1 - 11.4iT - 61T^{2} \)
67 \( 1 - 0.637T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 - 12.9iT - 79T^{2} \)
83 \( 1 + 2.55iT - 83T^{2} \)
89 \( 1 + 8.07iT - 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−10.25074648876128506338318681721, −9.550676430648308762286169190879, −8.808247989879596819644513940017, −7.996324193355279190012116764714, −7.43869079321045783331052257345, −6.38118529594997405789439837455, −5.26317723141196636359986077314, −4.28001630756652400639363861617, −3.67043572251269333682776065686, −2.17785474335175354399586202509, 0.65699842347373064594832515871, 2.07927927740205993251735972787, 3.26246123006698212308922876971, 3.84177542318314967076989662205, 5.05505850809413179871334444707, 6.29580076463873872783088359299, 7.44945309763595206457063432710, 8.270895930831576591782341513018, 8.952985867640703521920679859015, 9.768182132371565627426767118583

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