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Properties

Label 2-840-840.149-c0-0-0
Degree $2$
Conductor $840$
Sign $-0.553 - 0.832i$
Analytic cond. $0.419214$
Root an. cond. $0.647467$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + i·5-s − 0.999·6-s + (0.5 + 0.866i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.866 − 1.5i)11-s + (−0.866 − 0.499i)12-s + 0.999i·14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.499i)18-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + i·5-s − 0.999·6-s + (0.5 + 0.866i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.866 − 1.5i)11-s + (−0.866 − 0.499i)12-s + 0.999i·14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.499i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.553 - 0.832i$
Analytic conductor: \(0.419214\)
Root analytic conductor: \(0.647467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :0),\ -0.553 - 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.226880868\)
\(L(\frac12)\) \(\approx\) \(1.226880868\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 - iT \)
7 \( 1 + (-0.5 - 0.866i)T \)
good11 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.73T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + iT - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−10.92578912440746209165040910689, −10.20669707049911836799814929812, −8.819226255289743855170805200381, −8.017952097790370412944224225095, −6.97592437338449336494478054137, −5.98017812435011952422201262823, −5.66872359427170915677676297411, −4.63168311041163480254292357330, −3.44934377713721958884868886211, −2.55997465847858896904390362202, 1.15646066988147623267723523311, 2.22803853995054635380144638089, 4.09434557297842729133845139383, 4.89227480607770239440677642954, 5.26073125430857730685063985991, 6.58799718085550694940548408463, 7.31910880288427131656289753933, 8.203834933812492523209680584822, 9.717097593274354775532540792050, 10.34799317238006588007259229462

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