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Properties

Label 2-5040-5.4-c1-0-21
Degree $2$
Conductor $5040$
Sign $0.241 - 0.970i$
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  + (−2.17 − 0.539i)5-s + i·7-s + 2·11-s + 0.921i·13-s + 1.07i·17-s + 3.07·19-s + 2.34i·23-s + (4.41 + 2.34i)25-s − 6.68·29-s + 7.75·31-s + (0.539 − 2.17i)35-s − 10.8i·37-s − 6.49·41-s + 6.52i·43-s − 4.68i·47-s + ⋯
L(s)  = 1  + (−0.970 − 0.241i)5-s + 0.377i·7-s + 0.603·11-s + 0.255i·13-s + 0.261i·17-s + 0.706·19-s + 0.487i·23-s + (0.883 + 0.468i)25-s − 1.24·29-s + 1.39·31-s + (0.0911 − 0.366i)35-s − 1.78i·37-s − 1.01·41-s + 0.994i·43-s − 0.682i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.241 - 0.970i$
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: \((2,\ 5040,\ (\ :1/2),\ 0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.245760537\)
\(L(\frac12)\) \(\approx\) \(1.245760537\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 + 0.539i)T \)
7 \( 1 - iT \)
good11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 0.921iT - 13T^{2} \)
17 \( 1 - 1.07iT - 17T^{2} \)
19 \( 1 - 3.07T + 19T^{2} \)
23 \( 1 - 2.34iT - 23T^{2} \)
29 \( 1 + 6.68T + 29T^{2} \)
31 \( 1 - 7.75T + 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 + 6.49T + 41T^{2} \)
43 \( 1 - 6.52iT - 43T^{2} \)
47 \( 1 + 4.68iT - 47T^{2} \)
53 \( 1 + 3.75iT - 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 + 4.15T + 61T^{2} \)
67 \( 1 - 4.68iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 7.07iT - 73T^{2} \)
79 \( 1 - 6.15T + 79T^{2} \)
83 \( 1 - 6.83iT - 83T^{2} \)
89 \( 1 - 8.34T + 89T^{2} \)
97 \( 1 - 8.43iT - 97T^{2} \)
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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

−8.318643964019380009185737171785, −7.71101743766087116028270693545, −7.05603038644151751343865755104, −6.26107363175128710448672421915, −5.42290795547990417495948784424, −4.66697081590673390760875311394, −3.83689775190262803977540072414, −3.26206146346076700682016476130, −2.06354088447491457788936659538, −0.959680054145853547646999213604, 0.41805010445447578540211059537, 1.54388480140376012667139944279, 2.94718596344429781847333774117, 3.46613151040677573753265618692, 4.40655214554677806385565351258, 4.93264773155575801466974749638, 6.07496244670181324490394810395, 6.72683239428246625850110867421, 7.43098004551182944113014416273, 7.983832912667922165013991995172

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