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Properties

Label 2-840-35.33-c1-0-1
Degree $2$
Conductor $840$
Sign $-0.382 - 0.923i$
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  + (−0.965 − 0.258i)3-s + (2.12 + 0.695i)5-s + (−2.49 + 0.886i)7-s + (0.866 + 0.499i)9-s + (0.665 + 1.15i)11-s + (−2.75 + 2.75i)13-s + (−1.87 − 1.22i)15-s + (0.667 − 2.48i)17-s + (−2.03 + 3.53i)19-s + (2.63 − 0.211i)21-s + (−6.03 + 1.61i)23-s + (4.03 + 2.95i)25-s + (−0.707 − 0.707i)27-s + 0.00959i·29-s + (−0.377 + 0.217i)31-s + ⋯
L(s)  = 1  + (−0.557 − 0.149i)3-s + (0.950 + 0.310i)5-s + (−0.942 + 0.335i)7-s + (0.288 + 0.166i)9-s + (0.200 + 0.347i)11-s + (−0.764 + 0.764i)13-s + (−0.483 − 0.315i)15-s + (0.161 − 0.603i)17-s + (−0.467 + 0.809i)19-s + (0.575 − 0.0461i)21-s + (−1.25 + 0.336i)23-s + (0.806 + 0.590i)25-s + (−0.136 − 0.136i)27-s + 0.00178i·29-s + (−0.0677 + 0.0390i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487234 + 0.729119i\)
\(L(\frac12)\) \(\approx\) \(0.487234 + 0.729119i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (-2.12 - 0.695i)T \)
7 \( 1 + (2.49 - 0.886i)T \)
good11 \( 1 + (-0.665 - 1.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.75 - 2.75i)T - 13iT^{2} \)
17 \( 1 + (-0.667 + 2.48i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.03 - 3.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.03 - 1.61i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.00959iT - 29T^{2} \)
31 \( 1 + (0.377 - 0.217i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0737 + 0.275i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 + (-0.00860 - 0.00860i)T + 43iT^{2} \)
47 \( 1 + (4.20 - 1.12i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.26 - 8.45i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.24 - 7.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.42 + 3.70i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.46 - 1.99i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.71T + 71T^{2} \)
73 \( 1 + (7.35 + 1.97i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.941 - 0.543i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.94 - 4.94i)T - 83iT^{2} \)
89 \( 1 + (7.48 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.78 - 6.78i)T + 97iT^{2} \)
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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

−10.16858471459721620798543135775, −9.800307862792359954940000543651, −9.086926145405018278141031823262, −7.74400453301101898095075172738, −6.75200525709809723380994986474, −6.20933838993906236415023890387, −5.37791999780616040450924871471, −4.22661857335320929983073908246, −2.82641900185890674678185932374, −1.74449981218013221820791447353, 0.43637984537062521788239431123, 2.16500521412529357140180853233, 3.47897666344405319933111723000, 4.66637575935072375430884167283, 5.67197503137234615200908845711, 6.29304806736505971918882938791, 7.14583066163843886709774361323, 8.371632625289657315417478536943, 9.283788408847274583336783947929, 10.13987049998305154717640390534

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