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Properties

Label 2-4320-5.4-c1-0-45
Degree $2$
Conductor $4320$
Sign $0.998 + 0.0552i$
Analytic cond. $34.4953$
Root an. cond. $5.87327$
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 − 0.123i)5-s + 2.78i·7-s + 3.56·11-s + 0.895i·13-s − 6.09i·17-s − 4.09·19-s − 1.24i·23-s + (4.96 + 0.551i)25-s + 5.45·29-s − 5.84·31-s + (0.344 − 6.21i)35-s − 5.56i·37-s − 1.68·41-s − 3.47i·43-s + 11.6i·47-s + ⋯
L(s)  = 1  + (−0.998 − 0.0552i)5-s + 1.05i·7-s + 1.07·11-s + 0.248i·13-s − 1.47i·17-s − 0.939·19-s − 0.260i·23-s + (0.993 + 0.110i)25-s + 1.01·29-s − 1.04·31-s + (0.0581 − 1.05i)35-s − 0.915i·37-s − 0.262·41-s − 0.529i·43-s + 1.70i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4320\)    =    \(2^{5} \cdot 3^{3} \cdot 5\)
Sign: $0.998 + 0.0552i$
Analytic conductor: \(34.4953\)
Root analytic conductor: \(5.87327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4320} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4320,\ (\ :1/2),\ 0.998 + 0.0552i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.451736632\)
\(L(\frac12)\) \(\approx\) \(1.451736632\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.123i)T \)
good7 \( 1 - 2.78iT - 7T^{2} \)
11 \( 1 - 3.56T + 11T^{2} \)
13 \( 1 - 0.895iT - 13T^{2} \)
17 \( 1 + 6.09iT - 17T^{2} \)
19 \( 1 + 4.09T + 19T^{2} \)
23 \( 1 + 1.24iT - 23T^{2} \)
29 \( 1 - 5.45T + 29T^{2} \)
31 \( 1 + 5.84T + 31T^{2} \)
37 \( 1 + 5.56iT - 37T^{2} \)
41 \( 1 + 1.68T + 41T^{2} \)
43 \( 1 + 3.47iT - 43T^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 + 3.84iT - 53T^{2} \)
59 \( 1 - 9.82T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 7.13iT - 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 - 7.34iT - 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 + 7.68iT - 83T^{2} \)
89 \( 1 - 3.25T + 89T^{2} \)
97 \( 1 + 15.6iT - 97T^{2} \)
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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

−8.470744831325117559894858557461, −7.66177316581845722598862753131, −6.88835492799775136898370161734, −6.31816299114242515394124965619, −5.31859320781473869258980100085, −4.58923277146385315738770503831, −3.84088782434404805201799645847, −2.93881292913314467638178765272, −2.02969202767681957647817015344, −0.62502741323090490033321750045, 0.74984153499097313140319729578, 1.78371255408146759120953613380, 3.22772229210075547757434619908, 3.98025348183965475156223397723, 4.26062320846834906696431709952, 5.36903564270062055702997434141, 6.62010832992702443441971039189, 6.70468451193101384668956923002, 7.77404056091710849011112633312, 8.271851647583746538850436884125

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