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Properties

Label 2-840-840.797-c0-0-0
Degree $2$
Conductor $840$
Sign $-0.973 - 0.229i$
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 + (−0.707 − 0.707i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s − 1.00i·6-s − 7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s − 1.00·10-s − 1.41·11-s + (0.707 − 0.707i)12-s + (−0.707 − 0.707i)14-s + 1.00·15-s − 1.00·16-s + (−0.707 + 0.707i)18-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s − 1.00i·6-s − 7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s − 1.00·10-s − 1.41·11-s + (0.707 − 0.707i)12-s + (−0.707 − 0.707i)14-s + 1.00·15-s − 1.00·16-s + (−0.707 + 0.707i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.973 - 0.229i$
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: \((2,\ 840,\ (\ :0),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4893216119\)
\(L(\frac12)\) \(\approx\) \(0.4893216119\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + T \)
good11 \( 1 + 1.41T + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 - 2iT - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−10.84333654513157649117209871263, −10.31346855655920316388020058131, −8.737265721728093675809297799332, −7.86683003897690090601199294223, −7.09270131937858757513281960728, −6.64968749681639599445404503290, −5.62830624180351335834669574888, −4.84085991363408828379836606741, −3.44883219960331752616389779303, −2.62965208569480693567070984566, 0.40344781894006630169091161505, 2.64972558785853075190549312877, 3.78708303740029370136969056297, 4.46946809754759321338752913949, 5.46571477150493582131863355887, 6.05283097659108734459105842732, 7.31188919352870209698445981008, 8.544238871614343331613677014055, 9.696795315254791561325521185188, 9.979339573858511295645531491689

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