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Properties

Label 2-840-105.59-c1-0-5
Degree $2$
Conductor $840$
Sign $-0.695 + 0.718i$
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  + (−0.587 + 1.62i)3-s + (−2.19 + 0.427i)5-s + (−0.571 + 2.58i)7-s + (−2.30 − 1.91i)9-s + (4.72 + 2.72i)11-s − 5.23·13-s + (0.594 − 3.82i)15-s + (2.41 + 1.39i)17-s + (−2.40 + 1.38i)19-s + (−3.87 − 2.44i)21-s + (−3.63 − 6.28i)23-s + (4.63 − 1.87i)25-s + (4.47 − 2.63i)27-s + 1.09i·29-s + (−7.56 − 4.36i)31-s + ⋯
L(s)  = 1  + (−0.339 + 0.940i)3-s + (−0.981 + 0.191i)5-s + (−0.215 + 0.976i)7-s + (−0.769 − 0.638i)9-s + (1.42 + 0.822i)11-s − 1.45·13-s + (0.153 − 0.988i)15-s + (0.586 + 0.338i)17-s + (−0.551 + 0.318i)19-s + (−0.845 − 0.534i)21-s + (−0.757 − 1.31i)23-s + (0.926 − 0.375i)25-s + (0.861 − 0.507i)27-s + 0.202i·29-s + (−1.35 − 0.784i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.695 + 0.718i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112428 - 0.265059i\)
\(L(\frac12)\) \(\approx\) \(0.112428 - 0.265059i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.587 - 1.62i)T \)
5 \( 1 + (2.19 - 0.427i)T \)
7 \( 1 + (0.571 - 2.58i)T \)
good11 \( 1 + (-4.72 - 2.72i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.23T + 13T^{2} \)
17 \( 1 + (-2.41 - 1.39i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.40 - 1.38i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.63 + 6.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.09iT - 29T^{2} \)
31 \( 1 + (7.56 + 4.36i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.14 - 2.39i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.29T + 41T^{2} \)
43 \( 1 + 5.55iT - 43T^{2} \)
47 \( 1 + (-7.45 + 4.30i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.859 + 1.48i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.42 - 2.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.15 - 1.24i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.75 + 1.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.47iT - 71T^{2} \)
73 \( 1 + (1.39 - 2.41i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.59 - 7.96i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.588iT - 83T^{2} \)
89 \( 1 + (5.00 + 8.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−10.61894016268556496094203994516, −9.904431262494588583704543066594, −9.118642013657094384963853642565, −8.410522676459462279638643375004, −7.23047685424712779947968455176, −6.38292286591353709201973567731, −5.32039932571290883013768113378, −4.34900996730983733177546232631, −3.66694584323183438937692952499, −2.32743452610454428010610390950, 0.15113727383370575120774900160, 1.43181388560193851562126995179, 3.18731765748956377205665177273, 4.12036562051881766391815740709, 5.26104672267707969970298244551, 6.39933495344544492538127608451, 7.30014629398748162900778291978, 7.59992042008501834400543230183, 8.712050872738629964815345142976, 9.599313133923642734663614881376

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