Properties

Label 2-840-7.2-c1-0-13
Degree $2$
Conductor $840$
Sign $-0.469 + 0.883i$
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  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.0741 + 2.64i)7-s + (−0.499 + 0.866i)9-s + (−3.08 − 5.33i)11-s + 2.50·13-s − 0.999·15-s + (−2.90 − 5.02i)17-s + (−0.5 + 0.866i)19-s + (2.32 − 1.25i)21-s + (0.425 − 0.737i)23-s + (−0.499 − 0.866i)25-s + 0.999·27-s + 5.50·29-s + (−3.67 − 6.37i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (−0.0280 + 0.999i)7-s + (−0.166 + 0.288i)9-s + (−0.928 − 1.60i)11-s + 0.695·13-s − 0.258·15-s + (−0.703 − 1.21i)17-s + (−0.114 + 0.198i)19-s + (0.507 − 0.274i)21-s + (0.0887 − 0.153i)23-s + (−0.0999 − 0.173i)25-s + 0.192·27-s + 1.02·29-s + (−0.660 − 1.14i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.469 + 0.883i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.469 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.529099 - 0.880126i\)
\(L(\frac12)\) \(\approx\) \(0.529099 - 0.880126i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.0741 - 2.64i)T \)
good11 \( 1 + (3.08 + 5.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.50T + 13T^{2} \)
17 \( 1 + (2.90 + 5.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.425 + 0.737i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.50T + 29T^{2} \)
31 \( 1 + (3.67 + 6.37i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.90 + 8.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.14T + 41T^{2} \)
43 \( 1 + 7.16T + 43T^{2} \)
47 \( 1 + (-1.17 + 2.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.672 - 1.16i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.24 + 5.62i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.58 - 6.20i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.1T + 71T^{2} \)
73 \( 1 + (2.92 + 5.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.33 - 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.50T + 83T^{2} \)
89 \( 1 + (-5.40 + 9.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−9.884296910947715990452560555779, −8.776794821716763766482134947628, −8.475492468557794372022239176493, −7.39597993238665228804114073398, −6.14026224447207008166748614200, −5.73061952808094845781312368901, −4.77739823862750259314047641792, −3.19004792999902980564227393907, −2.20754931358730762982182936348, −0.51196472929046789237511805617, 1.67804961283783817152710506802, 3.15468598154034647288242839700, 4.30415156761521899372693141184, 4.94326358015456557952512389600, 6.26698076130004514229605673545, 6.94014662590591636527950218839, 7.87543556066126643044342158355, 8.847913733612866173959342935255, 10.06245646639645921346065342069, 10.32267483006597696668191827759

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