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Properties

Label 2-483-69.68-c1-0-8
Degree $2$
Conductor $483$
Sign $-0.729 - 0.683i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
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  + 0.800i·2-s + (−1.73 + 0.0370i)3-s + 1.35·4-s − 1.73·5-s + (−0.0296 − 1.38i)6-s + i·7-s + 2.68i·8-s + (2.99 − 0.128i)9-s − 1.38i·10-s + 1.58·11-s + (−2.35 + 0.0504i)12-s + 0.447·13-s − 0.800·14-s + (3.00 − 0.0644i)15-s + 0.569·16-s − 6.24·17-s + ⋯
L(s)  = 1  + 0.565i·2-s + (−0.999 + 0.0214i)3-s + 0.679·4-s − 0.776·5-s + (−0.0121 − 0.565i)6-s + 0.377i·7-s + 0.950i·8-s + (0.999 − 0.0428i)9-s − 0.439i·10-s + 0.479·11-s + (−0.679 + 0.0145i)12-s + 0.124·13-s − 0.213·14-s + (0.776 − 0.0166i)15-s + 0.142·16-s − 1.51·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.729 - 0.683i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.729 - 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.311752 + 0.788436i\)
\(L(\frac12)\) \(\approx\) \(0.311752 + 0.788436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 - 0.0370i)T \)
7 \( 1 - iT \)
23 \( 1 + (3.35 - 3.42i)T \)
good2 \( 1 - 0.800iT - 2T^{2} \)
5 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 - 1.58T + 11T^{2} \)
13 \( 1 - 0.447T + 13T^{2} \)
17 \( 1 + 6.24T + 17T^{2} \)
19 \( 1 - 6.45iT - 19T^{2} \)
29 \( 1 - 5.04iT - 29T^{2} \)
31 \( 1 - 4.59T + 31T^{2} \)
37 \( 1 + 4.25iT - 37T^{2} \)
41 \( 1 - 10.5iT - 41T^{2} \)
43 \( 1 - 1.24iT - 43T^{2} \)
47 \( 1 + 1.56iT - 47T^{2} \)
53 \( 1 + 1.95T + 53T^{2} \)
59 \( 1 + 13.0iT - 59T^{2} \)
61 \( 1 - 8.71iT - 61T^{2} \)
67 \( 1 + 9.90iT - 67T^{2} \)
71 \( 1 + 4.83iT - 71T^{2} \)
73 \( 1 - 2.41T + 73T^{2} \)
79 \( 1 - 12.8iT - 79T^{2} \)
83 \( 1 - 17.4T + 83T^{2} \)
89 \( 1 - 9.75T + 89T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−11.42344185049161298717314667692, −10.72211840920673137495147644096, −9.615112733345874861086267512375, −8.311348061448742901838116911397, −7.56329994172229211560322339713, −6.53697843990228650479079831156, −6.00718574527809375023700560571, −4.84757624770847928279531229459, −3.68631390543353600720878753004, −1.81909902733529833145278930701, 0.56705355330353260908823316230, 2.26338244042772441502497855767, 3.89168194835182802177681203251, 4.61893913471320902557342038496, 6.22917719827695225528164299545, 6.80910553299243045771889258173, 7.67317587550722043342431671087, 9.014945767924376950979240458903, 10.19369846865679512964741423310, 10.83991322146522182861226601946

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