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Properties

Label 2-840-7.2-c1-0-6
Degree $2$
Conductor $840$
Sign $0.266 - 0.963i$
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 + (0.5 + 2.59i)7-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)11-s + 13-s + 0.999·15-s + (−1.5 + 2.59i)19-s + (−2 + 1.73i)21-s + (−0.499 − 0.866i)25-s − 0.999·27-s + (4.5 + 7.79i)31-s + (−0.999 + 1.73i)33-s + (2.5 + 0.866i)35-s + (−1.5 + 2.59i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (0.188 + 0.981i)7-s + (−0.166 + 0.288i)9-s + (0.301 + 0.522i)11-s + 0.277·13-s + 0.258·15-s + (−0.344 + 0.596i)19-s + (−0.436 + 0.377i)21-s + (−0.0999 − 0.173i)25-s − 0.192·27-s + (0.808 + 1.39i)31-s + (−0.174 + 0.301i)33-s + (0.422 + 0.146i)35-s + (−0.246 + 0.427i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.266 - 0.963i$
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: \((2,\ 840,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39708 + 1.06284i\)
\(L(\frac12)\) \(\approx\) \(1.39708 + 1.06284i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-4.5 - 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 3T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-2 + 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−10.22019709268751083827880716406, −9.489243705300167228768064932667, −8.697066346978432145602927002099, −8.166857960707656671828415245793, −6.88369522405238895798616362975, −5.85877863001689386543455020266, −5.03835967732958229763471344174, −4.10131931787088016906218726859, −2.85523675790088000862321344173, −1.67580106651439227980727865016, 0.885142405585729386763805918767, 2.33028915159604487607400849703, 3.52544644542460809925575638554, 4.47718690693141757921533501906, 5.86549428400945092516621285032, 6.64003396140062945729501312863, 7.45247148479265075617247747637, 8.218959868804291794708721684218, 9.165789394297621450535099405290, 10.03771158579814575814368364027

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