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Properties

Label 2-840-280.139-c1-0-59
Degree $2$
Conductor $840$
Sign $0.411 + 0.911i$
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  + (−1.21 − 0.722i)2-s + 3-s + (0.955 + 1.75i)4-s + (2.09 − 0.785i)5-s + (−1.21 − 0.722i)6-s + (−2.63 + 0.274i)7-s + (0.108 − 2.82i)8-s + 9-s + (−3.11 − 0.557i)10-s + 1.22·11-s + (0.955 + 1.75i)12-s − 1.25i·13-s + (3.39 + 1.56i)14-s + (2.09 − 0.785i)15-s + (−2.17 + 3.35i)16-s + 3.68·17-s + ⋯
L(s)  = 1  + (−0.859 − 0.511i)2-s + 0.577·3-s + (0.477 + 0.878i)4-s + (0.936 − 0.351i)5-s + (−0.496 − 0.295i)6-s + (−0.994 + 0.103i)7-s + (0.0385 − 0.999i)8-s + 0.333·9-s + (−0.984 − 0.176i)10-s + 0.369·11-s + (0.275 + 0.507i)12-s − 0.349i·13-s + (0.907 + 0.419i)14-s + (0.540 − 0.202i)15-s + (−0.543 + 0.839i)16-s + 0.893·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.411 + 0.911i$
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: \((2,\ 840,\ (\ :1/2),\ 0.411 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16649 - 0.752822i\)
\(L(\frac12)\) \(\approx\) \(1.16649 - 0.752822i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 + 0.722i)T \)
3 \( 1 - T \)
5 \( 1 + (-2.09 + 0.785i)T \)
7 \( 1 + (2.63 - 0.274i)T \)
good11 \( 1 - 1.22T + 11T^{2} \)
13 \( 1 + 1.25iT - 13T^{2} \)
17 \( 1 - 3.68T + 17T^{2} \)
19 \( 1 + 6.93iT - 19T^{2} \)
23 \( 1 + 2.42T + 23T^{2} \)
29 \( 1 + 4.12iT - 29T^{2} \)
31 \( 1 - 2.80T + 31T^{2} \)
37 \( 1 - 6.71T + 37T^{2} \)
41 \( 1 - 11.6iT - 41T^{2} \)
43 \( 1 + 6.00iT - 43T^{2} \)
47 \( 1 - 1.84iT - 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 - 5.67iT - 59T^{2} \)
61 \( 1 + 6.48T + 61T^{2} \)
67 \( 1 - 5.15iT - 67T^{2} \)
71 \( 1 + 9.54iT - 71T^{2} \)
73 \( 1 + 1.61T + 73T^{2} \)
79 \( 1 + 8.33iT - 79T^{2} \)
83 \( 1 + 0.371T + 83T^{2} \)
89 \( 1 + 4.80iT - 89T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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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

−9.839890807075733498475124416693, −9.345873751562849823400832856473, −8.662175942173805903287446980561, −7.69456764682706706544864676252, −6.71345674158713417161616491552, −5.91178403399774617207521860103, −4.40367067283718698864688360709, −3.11870672644897660731858060436, −2.40477977403532775280533899294, −0.935271280229053335043946328275, 1.41149450823956721360034190025, 2.59893576480408210327425323467, 3.79261986801330025191389226156, 5.49404674352730302157204714845, 6.19791643160235414671279299515, 6.95122379042370882378929121262, 7.82962361183760552845518737750, 8.796725793562617541456517775957, 9.582372800742137620106337622319, 10.01887670385087315738025422295

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