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Properties

Label 2-840-280.139-c1-0-1
Degree $2$
Conductor $840$
Sign $-0.627 - 0.778i$
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  + (−0.911 − 1.08i)2-s + 3-s + (−0.339 + 1.97i)4-s + (−2.20 − 0.349i)5-s + (−0.911 − 1.08i)6-s + (0.795 + 2.52i)7-s + (2.44 − 1.42i)8-s + 9-s + (1.63 + 2.70i)10-s − 1.30·11-s + (−0.339 + 1.97i)12-s + 1.98i·13-s + (2.00 − 3.15i)14-s + (−2.20 − 0.349i)15-s + (−3.76 − 1.33i)16-s − 5.61·17-s + ⋯
L(s)  = 1  + (−0.644 − 0.764i)2-s + 0.577·3-s + (−0.169 + 0.985i)4-s + (−0.987 − 0.156i)5-s + (−0.371 − 0.441i)6-s + (0.300 + 0.953i)7-s + (0.863 − 0.505i)8-s + 0.333·9-s + (0.516 + 0.856i)10-s − 0.392·11-s + (−0.0980 + 0.568i)12-s + 0.549i·13-s + (0.535 − 0.844i)14-s + (−0.570 − 0.0903i)15-s + (−0.942 − 0.334i)16-s − 1.36·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.627 - 0.778i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.627 - 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0960990 + 0.200755i\)
\(L(\frac12)\) \(\approx\) \(0.0960990 + 0.200755i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.911 + 1.08i)T \)
3 \( 1 - T \)
5 \( 1 + (2.20 + 0.349i)T \)
7 \( 1 + (-0.795 - 2.52i)T \)
good11 \( 1 + 1.30T + 11T^{2} \)
13 \( 1 - 1.98iT - 13T^{2} \)
17 \( 1 + 5.61T + 17T^{2} \)
19 \( 1 + 3.69iT - 19T^{2} \)
23 \( 1 + 7.46T + 23T^{2} \)
29 \( 1 + 6.71iT - 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 1.94T + 37T^{2} \)
41 \( 1 - 2.70iT - 41T^{2} \)
43 \( 1 - 5.78iT - 43T^{2} \)
47 \( 1 - 9.66iT - 47T^{2} \)
53 \( 1 + 1.66T + 53T^{2} \)
59 \( 1 - 3.59iT - 59T^{2} \)
61 \( 1 + 5.08T + 61T^{2} \)
67 \( 1 + 2.27iT - 67T^{2} \)
71 \( 1 + 7.30iT - 71T^{2} \)
73 \( 1 + 8.44T + 73T^{2} \)
79 \( 1 + 7.73iT - 79T^{2} \)
83 \( 1 - 6.73T + 83T^{2} \)
89 \( 1 - 2.02iT - 89T^{2} \)
97 \( 1 + 1.76T + 97T^{2} \)
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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

−10.60686491259535331030875670913, −9.276140808250217715944163752115, −9.058603153325635418144829929298, −8.041714114064515763443600106854, −7.61686522466949633858902203350, −6.37755080687038536340342513631, −4.72318017136274671244580392186, −4.03792899275206624559536033723, −2.80223840266936160584959350903, −1.93017499452181332739452117892, 0.11950415412833510188917748385, 1.87694317187425773524397024287, 3.62649042442807956232695209347, 4.42248550395726387508702325082, 5.57928123929738745315829368645, 6.91760790254534910631406623033, 7.38578871827236252056641360568, 8.197022279380618908935958251824, 8.697360075302802104375006042678, 9.864575491220822169575854298152

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