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Properties

Label 2-840-840.773-c0-0-1
Degree $2$
Conductor $840$
Sign $0.164 - 0.986i$
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  + (0.965 + 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.965 + 0.258i)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 + (−0.258 + 0.448i)11-s + (−0.258 + 0.965i)12-s + (0.707 − 0.707i)14-s + (−0.499 − 0.866i)15-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)18-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.965 + 0.258i)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 + (−0.258 + 0.448i)11-s + (−0.258 + 0.965i)12-s + (0.707 − 0.707i)14-s + (−0.499 − 0.866i)15-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.164 - 0.986i$
Analytic conductor: \(0.419214\)
Root analytic conductor: \(0.647467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :0),\ 0.164 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.647666135\)
\(L(\frac12)\) \(\approx\) \(1.647666135\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good11 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 1.93iT - T^{2} \)
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 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
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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

−10.69916914355209797751530330221, −10.06919000154212433562038641090, −8.718645772528438496688322476805, −7.76073049756872905902704084333, −7.37693725216937370276598021906, −6.08531365463187779271357229501, −4.94111491268726452305213342695, −4.22732760666709855452440191300, −3.65657888287627876490924177171, −2.43560935776946512024050158955, 1.48934516220915141740543596101, 2.76696033461167667214449927955, 3.63230929198661244930268207917, 4.97747186390815855031335050747, 5.66840597260613660629097447842, 6.78483656586702344624562910717, 7.52739399573850347652931593596, 8.405452968026659124289049992743, 9.106375765745851332349257291842, 10.69884312512687721283855453045

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