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Properties

Label 2-840-840.437-c0-0-3
Degree $2$
Conductor $840$
Sign $-0.588 + 0.808i$
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 + (−0.258 − 0.965i)5-s − i·6-s + (0.5 − 0.866i)7-s + (−0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s − 10-s + (−0.965 + 1.67i)11-s + (−0.965 − 0.258i)12-s + (−0.707 − 0.707i)14-s + (−0.499 − 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 + (−0.258 − 0.965i)5-s − i·6-s + (0.5 − 0.866i)7-s + (−0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s − 10-s + (−0.965 + 1.67i)11-s + (−0.965 − 0.258i)12-s + (−0.707 − 0.707i)14-s + (−0.499 − 0.866i)15-s + (0.500 + 0.866i)16-s + (−0.258 − 0.965i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.588 + 0.808i$
Analytic conductor: \(0.419214\)
Root analytic conductor: \(0.647467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :0),\ -0.588 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329503355\)
\(L(\frac12)\) \(\approx\) \(1.329503355\)
\(L(1)\) 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 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good11 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 0.517iT - T^{2} \)
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 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
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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

−9.996044684719772352726988051089, −9.493884411798251862296563850332, −8.409319444573294513233752894308, −7.87334962374788100581439185545, −6.91236378399525181181086642187, −5.10194172408825982133529397407, −4.56437804683084549198181442219, −3.69017483547221370287753208490, −2.33467745820552917448857059198, −1.34299116894686525499061135156, 2.63201939088498195861021535474, 3.28598110981237243023686356723, 4.44219810274191962735265831136, 5.57846486445492022768891685485, 6.32705156077719200858507225804, 7.55063265136791578753685568353, 8.102793646141710567114324243247, 8.668910249342070594024220484462, 9.615212266857280859748667198670, 10.60584311204985918307633005411

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