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Properties

Label 2-840-280.139-c1-0-57
Degree $2$
Conductor $840$
Sign $0.916 + 0.399i$
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  + (0.965 − 1.03i)2-s + 3-s + (−0.135 − 1.99i)4-s + (0.836 + 2.07i)5-s + (0.965 − 1.03i)6-s + (−0.292 + 2.62i)7-s + (−2.19 − 1.78i)8-s + 9-s + (2.95 + 1.13i)10-s + 4.26·11-s + (−0.135 − 1.99i)12-s + 0.507i·13-s + (2.43 + 2.84i)14-s + (0.836 + 2.07i)15-s + (−3.96 + 0.539i)16-s + 2.21·17-s + ⋯
L(s)  = 1  + (0.682 − 0.730i)2-s + 0.577·3-s + (−0.0676 − 0.997i)4-s + (0.373 + 0.927i)5-s + (0.394 − 0.421i)6-s + (−0.110 + 0.993i)7-s + (−0.775 − 0.631i)8-s + 0.333·9-s + (0.932 + 0.360i)10-s + 1.28·11-s + (−0.0390 − 0.576i)12-s + 0.140i·13-s + (0.650 + 0.759i)14-s + (0.215 + 0.535i)15-s + (−0.990 + 0.134i)16-s + 0.536·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.916 + 0.399i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.916 + 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.86963 - 0.598770i\)
\(L(\frac12)\) \(\approx\) \(2.86963 - 0.598770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 1.03i)T \)
3 \( 1 - T \)
5 \( 1 + (-0.836 - 2.07i)T \)
7 \( 1 + (0.292 - 2.62i)T \)
good11 \( 1 - 4.26T + 11T^{2} \)
13 \( 1 - 0.507iT - 13T^{2} \)
17 \( 1 - 2.21T + 17T^{2} \)
19 \( 1 + 0.143iT - 19T^{2} \)
23 \( 1 + 1.99T + 23T^{2} \)
29 \( 1 + 1.26iT - 29T^{2} \)
31 \( 1 - 5.61T + 31T^{2} \)
37 \( 1 - 8.48T + 37T^{2} \)
41 \( 1 + 11.0iT - 41T^{2} \)
43 \( 1 + 5.73iT - 43T^{2} \)
47 \( 1 - 6.77iT - 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 - 5.17iT - 59T^{2} \)
61 \( 1 + 14.5T + 61T^{2} \)
67 \( 1 + 8.16iT - 67T^{2} \)
71 \( 1 - 4.59iT - 71T^{2} \)
73 \( 1 + 6.89T + 73T^{2} \)
79 \( 1 + 14.2iT - 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + 7.79iT - 89T^{2} \)
97 \( 1 - 7.43T + 97T^{2} \)
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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

−10.11807208754618908128404707918, −9.452510149795351526666789402962, −8.825578870207479533351433856242, −7.47285201281141904306240505546, −6.32389737412501864402681623597, −5.91113398659466807438578366888, −4.52112920804791502045413299571, −3.47934814559558422486104716472, −2.66469472638122974574704399596, −1.67314886480893324073581999903, 1.32565037733471104681974774215, 3.07221263399064898785897858032, 4.15031115071561739245878737310, 4.68097579631022962699710363227, 6.00411181565112269709092012544, 6.68519374071179963901900465774, 7.78791756212971141551145585202, 8.311355893868691331441282942881, 9.364600766874318421953905585511, 9.875934164866070063032756180588

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