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Properties

Label 2-840-280.139-c1-0-50
Degree $2$
Conductor $840$
Sign $0.861 + 0.507i$
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  + (−0.752 − 1.19i)2-s + 3-s + (−0.867 + 1.80i)4-s + (1.64 + 1.51i)5-s + (−0.752 − 1.19i)6-s + (2.63 − 0.259i)7-s + (2.81 − 0.318i)8-s + 9-s + (0.571 − 3.11i)10-s − 3.15·11-s + (−0.867 + 1.80i)12-s − 2.42i·13-s + (−2.29 − 2.95i)14-s + (1.64 + 1.51i)15-s + (−2.49 − 3.12i)16-s + 2.13·17-s + ⋯
L(s)  = 1  + (−0.532 − 0.846i)2-s + 0.577·3-s + (−0.433 + 0.901i)4-s + (0.736 + 0.676i)5-s + (−0.307 − 0.488i)6-s + (0.995 − 0.0980i)7-s + (0.993 − 0.112i)8-s + 0.333·9-s + (0.180 − 0.983i)10-s − 0.952·11-s + (−0.250 + 0.520i)12-s − 0.673i·13-s + (−0.612 − 0.790i)14-s + (0.425 + 0.390i)15-s + (−0.623 − 0.781i)16-s + 0.517·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.861 + 0.507i$
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: \((2,\ 840,\ (\ :1/2),\ 0.861 + 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69432 - 0.461594i\)
\(L(\frac12)\) \(\approx\) \(1.69432 - 0.461594i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.752 + 1.19i)T \)
3 \( 1 - T \)
5 \( 1 + (-1.64 - 1.51i)T \)
7 \( 1 + (-2.63 + 0.259i)T \)
good11 \( 1 + 3.15T + 11T^{2} \)
13 \( 1 + 2.42iT - 13T^{2} \)
17 \( 1 - 2.13T + 17T^{2} \)
19 \( 1 + 2.84iT - 19T^{2} \)
23 \( 1 - 7.13T + 23T^{2} \)
29 \( 1 - 4.57iT - 29T^{2} \)
31 \( 1 + 3.50T + 31T^{2} \)
37 \( 1 - 7.05T + 37T^{2} \)
41 \( 1 - 2.43iT - 41T^{2} \)
43 \( 1 + 5.07iT - 43T^{2} \)
47 \( 1 - 12.8iT - 47T^{2} \)
53 \( 1 - 4.91T + 53T^{2} \)
59 \( 1 + 2.93iT - 59T^{2} \)
61 \( 1 + 1.17T + 61T^{2} \)
67 \( 1 + 5.88iT - 67T^{2} \)
71 \( 1 - 5.30iT - 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 4.53iT - 79T^{2} \)
83 \( 1 + 14.0T + 83T^{2} \)
89 \( 1 + 4.75iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−10.24496935307986865746963333507, −9.356274540638160580366597018579, −8.599845306805222497827669197786, −7.67685178629659704908123602544, −7.15393938385361605686895981111, −5.52551199170563816387972636343, −4.63253269001727770967914260791, −3.16462193395160669094902364565, −2.56300339240086238097863792876, −1.32158507996242605339852719216, 1.24874125266203318935684414165, 2.33930119347875176785468373755, 4.27151437248069034111844800695, 5.15381178227680265847607455103, 5.80152873540413593478071339011, 7.06179743632334509051035092680, 7.88785177610908979406990703572, 8.529831722443634645154093890904, 9.219820678848225146296169706566, 10.00202415392566848175284338897

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