Properties

Label 2-840-840.293-c0-0-6
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 + (−0.923 + 0.382i)3-s + 1.00i·4-s + (0.382 − 0.923i)5-s + (−0.923 − 0.382i)6-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (−0.382 − 0.923i)12-s + (1.30 + 1.30i)13-s + 1.00·14-s + i·15-s − 1.00·16-s + 18-s − 0.765i·19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.923 + 0.382i)3-s + 1.00i·4-s + (0.382 − 0.923i)5-s + (−0.923 − 0.382i)6-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (−0.382 − 0.923i)12-s + (1.30 + 1.30i)13-s + 1.00·14-s + i·15-s − 1.00·16-s + 18-s − 0.765i·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: $\chi_{840} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.221486002\)
\(L(\frac12)\) \(\approx\) \(1.221486002\)
\(L(1)\) 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 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + T^{2} \)
13 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 0.765iT - T^{2} \)
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 \( 1 - 0.765T + T^{2} \)
61 \( 1 + 1.84T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
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

−10.79193730136006397299967453916, −9.547089552781756117844614625107, −8.828681562710880789735110086842, −7.86526385535305695713282273769, −6.82729055868270970397192204499, −6.06115572596507711562379276359, −5.22509151741996889788123356725, −4.39578220433916340209993048637, −3.84516639899013470651620436230, −1.58331240300568775852231981741, 1.51281901516547494797489065451, 2.61217345257461381476611424368, 3.85114181815973313376553693970, 5.09154197067881635994706699895, 5.99977438097242549376257624509, 6.19552889804234249517259592129, 7.57956017569464895337859526818, 8.575857172733136569761047441610, 9.987319846111477838669689583939, 10.55029568519144654848499301715

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