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Properties

Label 2-840-840.173-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(1.059930068\)
\(L(\frac12)\) \(\approx\) \(1.059930068\)
\(L(1)\) 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

−10.23749269295426264573631600528, −9.295266556631414362993233751509, −8.590942934034668938050297101317, −7.958903813334545559580106074565, −7.39362905389533254035703319626, −5.34364778561385694772436878923, −4.35736319958733769761809533418, −3.85004326128125025205705615501, −2.60601652398484044463543941972, −1.29503207103793846526258186431, 1.84781417752647886055064611567, 3.29954092534597862237446575698, 4.36892170439573638924391308350, 5.35272201888451707591365936104, 6.69205955316941513002216516300, 7.29863668427195134700624749964, 8.036041215203770324462237440464, 8.608132089101698266364903876110, 9.473437452868436082568724856840, 10.40861629590693227407129918853

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