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Properties

Label 2-840-840.173-c0-0-1
Degree $2$
Conductor $840$
Sign $-0.104 - 0.994i$
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  + (0.258 + 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + (0.707 + 0.707i)5-s − i·6-s + (0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (−0.500 + 0.866i)10-s + (0.258 + 0.448i)11-s + (0.965 − 0.258i)12-s + (0.707 + 0.707i)14-s + (−0.500 − 0.866i)15-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)18-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + (0.707 + 0.707i)5-s − i·6-s + (0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (−0.500 + 0.866i)10-s + (0.258 + 0.448i)11-s + (0.965 − 0.258i)12-s + (0.707 + 0.707i)14-s + (−0.500 − 0.866i)15-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.104 - 0.994i$
Analytic conductor: \(0.419214\)
Root analytic conductor: \(0.647467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :0),\ -0.104 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9243096120\)
\(L(\frac12)\) \(\approx\) \(0.9243096120\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
good11 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 - 1.93iT - T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
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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

−10.64289074311544043552846898701, −9.861532001930152436308014277019, −8.840895373304027469968089606704, −7.66134062137048024013354381924, −7.03388926310273633441392115163, −6.42532643343402039929156936192, −5.36376485570038066196808691500, −4.82797975782431146036849003870, −3.57087313774048751370641493442, −1.68110877530432933889416178521, 1.16124770820420169424709460264, 2.33168563745036726003788325163, 4.04002021387731848943520732532, 4.76895125185253623206462607289, 5.66735845569632521160431377743, 6.10634228676770709992982025430, 7.81987411470749374682335498946, 8.919071818653801822376943672967, 9.452210537628957051318787616167, 10.37604256845838570418068716009

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