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Properties

Label 2-840-35.12-c1-0-2
Degree $2$
Conductor $840$
Sign $-0.310 - 0.950i$
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  + (−0.258 + 0.965i)3-s + (−0.519 − 2.17i)5-s + (−2.60 + 0.467i)7-s + (−0.866 − 0.499i)9-s + (−0.472 − 0.819i)11-s + (2.05 + 2.05i)13-s + (2.23 + 0.0610i)15-s + (3.88 + 1.04i)17-s + (−2.94 + 5.10i)19-s + (0.222 − 2.63i)21-s + (1.68 + 6.27i)23-s + (−4.46 + 2.26i)25-s + (0.707 − 0.707i)27-s + 8.43i·29-s + (−9.18 + 5.30i)31-s + ⋯
L(s)  = 1  + (−0.149 + 0.557i)3-s + (−0.232 − 0.972i)5-s + (−0.984 + 0.176i)7-s + (−0.288 − 0.166i)9-s + (−0.142 − 0.246i)11-s + (0.570 + 0.570i)13-s + (0.577 + 0.0157i)15-s + (0.943 + 0.252i)17-s + (−0.676 + 1.17i)19-s + (0.0485 − 0.575i)21-s + (0.350 + 1.30i)23-s + (−0.892 + 0.452i)25-s + (0.136 − 0.136i)27-s + 1.56i·29-s + (−1.65 + 0.952i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.310 - 0.950i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (817, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.310 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.480233 + 0.661775i\)
\(L(\frac12)\) \(\approx\) \(0.480233 + 0.661775i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (0.519 + 2.17i)T \)
7 \( 1 + (2.60 - 0.467i)T \)
good11 \( 1 + (0.472 + 0.819i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.05 - 2.05i)T + 13iT^{2} \)
17 \( 1 + (-3.88 - 1.04i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.94 - 5.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.68 - 6.27i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 8.43iT - 29T^{2} \)
31 \( 1 + (9.18 - 5.30i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.79 + 1.55i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.222iT - 41T^{2} \)
43 \( 1 + (-4.06 + 4.06i)T - 43iT^{2} \)
47 \( 1 + (2.05 + 7.65i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (9.43 + 2.52i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-6.62 - 11.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.70 + 5.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.33 - 4.98i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.74T + 71T^{2} \)
73 \( 1 + (0.846 - 3.15i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.40 + 2.54i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.64 - 9.64i)T + 83iT^{2} \)
89 \( 1 + (6.58 - 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.60 - 4.60i)T - 97iT^{2} \)
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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

−10.40032015829962118964873033761, −9.454710850386162487546038135623, −8.973244834770916706549908643883, −8.095159490157189228468157575681, −6.99324058919334761292147859194, −5.80512301397932931512061488399, −5.31709939388014431512581306835, −3.93296571707616552814490622593, −3.39799744985567646143928272354, −1.48059397613237057493896566518, 0.41690508053348142612468376683, 2.44328744212154287799944661421, 3.23835003195861769068302852408, 4.43165587556821026140643943379, 5.97571363112810594474810405331, 6.39442164972780875571135362989, 7.38361437091845063711216930355, 7.961181978885783735561507882314, 9.220598407078051966926097412010, 10.01054332742225535635749039538

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