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Properties

Label 2-840-56.27-c1-0-42
Degree $2$
Conductor $840$
Sign $0.921 + 0.387i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + i·3-s − 2i·4-s + 5-s + (−1 − i)6-s + (−1 − 2.44i)7-s + (2 + 2i)8-s − 9-s + (−1 + i)10-s + 2·11-s + 2·12-s − 2.89·13-s + (3.44 + 1.44i)14-s + i·15-s − 4·16-s − 4.89i·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + 0.577i·3-s − i·4-s + 0.447·5-s + (−0.408 − 0.408i)6-s + (−0.377 − 0.925i)7-s + (0.707 + 0.707i)8-s − 0.333·9-s + (−0.316 + 0.316i)10-s + 0.603·11-s + 0.577·12-s − 0.804·13-s + (0.921 + 0.387i)14-s + 0.258i·15-s − 16-s − 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.921 + 0.387i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.921 + 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895276 - 0.180457i\)
\(L(\frac12)\) \(\approx\) \(0.895276 - 0.180457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 - iT \)
5 \( 1 - T \)
7 \( 1 + (1 + 2.44i)T \)
good11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 2.89T + 13T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 - 2.89iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 0.898iT - 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 11.7iT - 37T^{2} \)
41 \( 1 + 6.89iT - 41T^{2} \)
43 \( 1 - 0.898T + 43T^{2} \)
47 \( 1 - 1.10T + 47T^{2} \)
53 \( 1 + 9.79iT - 53T^{2} \)
59 \( 1 - 12.8iT - 59T^{2} \)
61 \( 1 - 8.89T + 61T^{2} \)
67 \( 1 - 4.89T + 67T^{2} \)
71 \( 1 - 3.10iT - 71T^{2} \)
73 \( 1 - 0.898iT - 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 10.8iT - 89T^{2} \)
97 \( 1 - 8.89iT - 97T^{2} \)
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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

−10.01334679790182874997890569062, −9.380059128025472333430717326864, −8.611314447227260605385258934628, −7.44749998112824114440881250954, −6.86872095485982432403570699224, −5.88691031075885493192251905792, −4.92343089475012949743248018228, −3.96956994945516123206626827927, −2.36396885729539513429862654446, −0.59119632415861262998445627595, 1.41726052762611646801032314501, 2.44206766401627630794556078687, 3.41564682262453430423449328815, 4.87519505601573175090450087739, 6.13249242941476659268998784643, 6.86423710454969580252818381229, 7.938703482161444650372782190128, 8.685888663202085524847203988647, 9.496535415856454750703669779110, 10.02543453668307822277404045297

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