Properties

Label 2-5040-5.4-c1-0-54
Degree $2$
Conductor $5040$
Sign $0.447 + 0.894i$
Analytic cond. $40.2446$
Root an. cond. $6.34386$
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  + (−2 + i)5-s i·7-s + 4·11-s − 2i·13-s − 2i·17-s − 2·19-s + 6i·23-s + (3 − 4i)25-s + 6·29-s − 6·31-s + (1 + 2i)35-s − 4i·37-s − 4i·43-s + 4i·47-s − 49-s + ⋯
L(s)  = 1  + (−0.894 + 0.447i)5-s − 0.377i·7-s + 1.20·11-s − 0.554i·13-s − 0.485i·17-s − 0.458·19-s + 1.25i·23-s + (0.600 − 0.800i)25-s + 1.11·29-s − 1.07·31-s + (0.169 + 0.338i)35-s − 0.657i·37-s − 0.609i·43-s + 0.583i·47-s − 0.142·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(40.2446\)
Root analytic conductor: \(6.34386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5040} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5040,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.356704333\)
\(L(\frac12)\) \(\approx\) \(1.356704333\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (2 - i)T \)
7 \( 1 + iT \)
good11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−7.913623775180740305041447614234, −7.44405477547172090249095034002, −6.76859367155054717686707958605, −6.08773216882187887150194051471, −5.10362541349360466206183278186, −4.22233960846044003554235007887, −3.63591265806906311126248800952, −2.91056700258392534705339202068, −1.62384918158256139771304552022, −0.45365333101092635130843334355, 0.953376686207962207846543021984, 1.98741969506551593509163250878, 3.14272197517303297972975012282, 4.03796273095759626209845775860, 4.48843887522150546993028800139, 5.36239487320947567970015956367, 6.54675306222340887426383423112, 6.64294914694382344016972255277, 7.81589703406006464654024524562, 8.383949834434229405597320807512

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