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Properties

Label 2-840-840.797-c0-0-5
Degree $2$
Conductor $840$
Sign $0.525 + 0.850i$
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  + (−0.707 + 0.707i)2-s + (−0.382 + 0.923i)3-s − 1.00i·4-s + (−0.923 + 0.382i)5-s + (−0.382 − 0.923i)6-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.382 − 0.923i)10-s + (0.923 + 0.382i)12-s + (−0.541 + 0.541i)13-s + 1.00·14-s − i·15-s − 1.00·16-s + 18-s − 1.84i·19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.382 + 0.923i)3-s − 1.00i·4-s + (−0.923 + 0.382i)5-s + (−0.382 − 0.923i)6-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.382 − 0.923i)10-s + (0.923 + 0.382i)12-s + (−0.541 + 0.541i)13-s + 1.00·14-s − i·15-s − 1.00·16-s + 18-s − 1.84i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(0.419214\)
Root analytic conductor: \(0.647467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :0),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1750864545\)
\(L(\frac12)\) \(\approx\) \(0.1750864545\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + T^{2} \)
13 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + 1.84iT - T^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + 1.84T + T^{2} \)
61 \( 1 + 0.765T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−10.22567940190814304387754338166, −9.418670159911763129464192980516, −8.716768499566425109567626270583, −7.62882154370374393745749445059, −6.83324538811742872264712317638, −6.20720151604004699083869255318, −4.79037348430907972409233403756, −4.22760948806766480507821910609, −2.86112427347970306147167813418, −0.23268649900585883291428216113, 1.57622486052582564016848014234, 2.88604215080063581166308535490, 3.88832938884724760510556981396, 5.35279101507200909719597279142, 6.34602570806304002307911038068, 7.56182946060276040619216196599, 7.913852965275893498573504110748, 8.774779484700329283491748978111, 9.745259742073124331476742790291, 10.56866998043723203331269948086

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