Properties

Label 2-5040-105.104-c1-0-37
Degree $2$
Conductor $5040$
Sign $0.975 - 0.220i$
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  + (−1.08 + 1.95i)5-s + (0.595 − 2.57i)7-s − 3.74i·11-s − 3.36·13-s + 0.841i·17-s + 5.59i·19-s + 2.35·23-s + (−2.64 − 4.24i)25-s − 1.41i·29-s + 8.66i·31-s + (4.39 + 3.96i)35-s + 5.15i·37-s − 5.74·41-s + 3.32i·43-s − 6.43i·47-s + ⋯
L(s)  = 1  + (−0.485 + 0.874i)5-s + (0.224 − 0.974i)7-s − 1.12i·11-s − 0.931·13-s + 0.204i·17-s + 1.28i·19-s + 0.490·23-s + (−0.529 − 0.848i)25-s − 0.262i·29-s + 1.55i·31-s + (0.742 + 0.669i)35-s + 0.847i·37-s − 0.896·41-s + 0.507i·43-s − 0.938i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.975 - 0.220i$
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: \((2,\ 5040,\ (\ :1/2),\ 0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.481478533\)
\(L(\frac12)\) \(\approx\) \(1.481478533\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 - 1.95i)T \)
7 \( 1 + (-0.595 + 2.57i)T \)
good11 \( 1 + 3.74iT - 11T^{2} \)
13 \( 1 + 3.36T + 13T^{2} \)
17 \( 1 - 0.841iT - 17T^{2} \)
19 \( 1 - 5.59iT - 19T^{2} \)
23 \( 1 - 2.35T + 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 - 5.15iT - 37T^{2} \)
41 \( 1 + 5.74T + 41T^{2} \)
43 \( 1 - 3.32iT - 43T^{2} \)
47 \( 1 + 6.43iT - 47T^{2} \)
53 \( 1 - 9.64T + 53T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 1.82iT - 67T^{2} \)
71 \( 1 + 3.74iT - 71T^{2} \)
73 \( 1 - 0.979T + 73T^{2} \)
79 \( 1 - 6.58T + 79T^{2} \)
83 \( 1 + 12.5iT - 83T^{2} \)
89 \( 1 - 2.16T + 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−8.215317766822188167646412762914, −7.47232771019331920112970070661, −6.92791498796392918266396840172, −6.25338524429212426612084400544, −5.33095019741909664675973956845, −4.48952135672373028158441534870, −3.54376767388471928342587955991, −3.19208604367914225317296360855, −1.94586511013226772732632293271, −0.69028748184536300786736275961, 0.61687414811875295512827235352, 2.02142800932232527357802051154, 2.59072938461076807842183407387, 3.85693063491561482853004022693, 4.74719666779967329030493056699, 5.06622711056058188562141128956, 5.87473145067909895986832889781, 7.05615547097083659778789965452, 7.39240474404925953555759168391, 8.301321344637490219328047038266

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