Properties

Label 2-5040-5.4-c1-0-60
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 − 6i·13-s + 2i·17-s + 6·19-s + 2i·23-s + (3 − 4i)25-s + 6·29-s + 2·31-s + (1 + 2i)35-s + 4i·37-s − 8·41-s + 4i·43-s − 4i·47-s + ⋯
L(s)  = 1  + (0.894 − 0.447i)5-s + 0.377i·7-s − 1.20·11-s − 1.66i·13-s + 0.485i·17-s + 1.37·19-s + 0.417i·23-s + (0.600 − 0.800i)25-s + 1.11·29-s + 0.359·31-s + (0.169 + 0.338i)35-s + 0.657i·37-s − 1.24·41-s + 0.609i·43-s − 0.583i·47-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

\(L(1)\) \(\approx\) \(2.089325744\)
\(L(\frac12)\) \(\approx\) \(2.089325744\)
\(L(\frac{3}{2})\) 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 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−8.233049229312998812374115683297, −7.54027432318861647411878068266, −6.55970969453133521166372124521, −5.69278549745144952098938841180, −5.32163474782719265398877749467, −4.73086495946849708430634590333, −3.24851573616521442795489052846, −2.81433610684498022434565858824, −1.71178716529098566143770020686, −0.61118958819002827813211708971, 1.08012451450206714257082681843, 2.20657164050761643819472646699, 2.82907069222483250583880732454, 3.84408942843230488118187314515, 4.88621676502434601966023664023, 5.33140779341250005998173094597, 6.34017352639495177583777399026, 6.89827415027531138137850571021, 7.49108617024118605402482255855, 8.371422190880133544810976538365

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