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Properties

Label 2-840-840.149-c0-0-3
Degree $2$
Conductor $840$
Sign $-0.444 + 0.895i$
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 + (−0.866 + 0.5i)5-s − 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + 0.999·10-s + (−0.866 − 1.5i)11-s + (0.866 + 0.499i)12-s + 0.999i·14-s + (−0.499 + 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 + (−0.866 + 0.5i)5-s − 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + 0.999·10-s + (−0.866 − 1.5i)11-s + (0.866 + 0.499i)12-s + 0.999i·14-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.444 + 0.895i$
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: \((2,\ 840,\ (\ :0),\ -0.444 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6428271796\)
\(L(\frac12)\) \(\approx\) \(0.6428271796\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
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

−10.24990375266783894208143728593, −9.193024435960010970810667163703, −8.276855337427240096621370564326, −7.86189964280162824692180604220, −7.04570626348119670308580300052, −6.25516660109818317283211771018, −4.15956083774867056343353984795, −3.29630953780711754438619651704, −2.67895887576816873589101813255, −0.78080496692233655192936998471, 2.00325116752297824852606799818, 3.09985963295911593686661986388, 4.62842248268027809634048145414, 5.22899022101729665070302856866, 6.72111745596697213266417962639, 7.53749228545587394862456934963, 8.293252034786180950949521422061, 8.853721284362381483867225216820, 9.734211731518223675409328245262, 10.22590914524953154160621070651

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