Properties

Label 2-5040-105.104-c1-0-42
Degree $2$
Conductor $5040$
Sign $0.0515 + 0.998i$
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  − 2.23·5-s + (−2.23 − 1.41i)7-s + 5.65i·11-s − 4.47·13-s + 3.16i·17-s + 3.16i·19-s − 4·23-s + 5.00·25-s − 2.82i·29-s + 6.32i·31-s + (5.00 + 3.16i)35-s − 9.89i·37-s − 4.47·41-s + 1.41i·43-s + 9.48i·47-s + ⋯
L(s)  = 1  − 0.999·5-s + (−0.845 − 0.534i)7-s + 1.70i·11-s − 1.24·13-s + 0.766i·17-s + 0.725i·19-s − 0.834·23-s + 1.00·25-s − 0.525i·29-s + 1.13i·31-s + (0.845 + 0.534i)35-s − 1.62i·37-s − 0.698·41-s + 0.215i·43-s + 1.38i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.0515 + 0.998i$
Analytic conductor: \(40.2446\)
Root analytic conductor: \(6.34386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5040} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5040,\ (\ :1/2),\ 0.0515 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2896433498\)
\(L(\frac12)\) \(\approx\) \(0.2896433498\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T \)
7 \( 1 + (2.23 + 1.41i)T \)
good11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 - 3.16iT - 17T^{2} \)
19 \( 1 - 3.16iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 - 6.32iT - 31T^{2} \)
37 \( 1 + 9.89iT - 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 - 1.41iT - 43T^{2} \)
47 \( 1 - 9.48iT - 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 + 9.48iT - 61T^{2} \)
67 \( 1 - 7.07iT - 67T^{2} \)
71 \( 1 - 1.41iT - 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + 12.6iT - 83T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−7.77003541645011171722771078274, −7.41816570338724371271272961094, −6.84947001116294025536688769246, −5.99311272480285650491998092623, −4.91462802490384412718353213946, −4.25640024554332225054050450321, −3.71253151417336640592208087826, −2.68053844332892998366971013893, −1.67829232253079414466782810524, −0.11998043303085641195770618318, 0.67000876403028149922132624699, 2.45379452252262860207170576032, 3.09164769135147385461561817574, 3.76537166971372340244400138171, 4.78123586625999750233090330331, 5.47677645822631811091583762449, 6.31022088027317081745240555654, 7.01466814274357750611833581427, 7.66445396549220074319729770430, 8.556175268292677744687583630011

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