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Properties

Label 2-483-161.160-c1-0-27
Degree $2$
Conductor $483$
Sign $0.918 + 0.395i$
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  + 2.51·2-s − i·3-s + 4.34·4-s − 1.21·5-s − 2.51i·6-s + (2.34 + 1.21i)7-s + 5.89·8-s − 9-s − 3.06·10-s − 1.13i·11-s − 4.34i·12-s + 0.518i·13-s + (5.91 + 3.06i)14-s + 1.21i·15-s + 6.16·16-s − 3.06·17-s + ⋯
L(s)  = 1  + 1.78·2-s − 0.577i·3-s + 2.17·4-s − 0.544·5-s − 1.02i·6-s + (0.887 + 0.460i)7-s + 2.08·8-s − 0.333·9-s − 0.969·10-s − 0.341i·11-s − 1.25i·12-s + 0.143i·13-s + (1.58 + 0.819i)14-s + 0.314i·15-s + 1.54·16-s − 0.743·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.918 + 0.395i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.918 + 0.395i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.63525 - 0.749911i\)
\(L(\frac12)\) \(\approx\) \(3.63525 - 0.749911i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (-2.34 - 1.21i)T \)
23 \( 1 + (0.341 + 4.78i)T \)
good2 \( 1 - 2.51T + 2T^{2} \)
5 \( 1 + 1.21T + 5T^{2} \)
11 \( 1 + 1.13iT - 11T^{2} \)
13 \( 1 - 0.518iT - 13T^{2} \)
17 \( 1 + 3.06T + 17T^{2} \)
19 \( 1 - 1.13T + 19T^{2} \)
29 \( 1 + 2.17T + 29T^{2} \)
31 \( 1 - 6.85iT - 31T^{2} \)
37 \( 1 - 10.1iT - 37T^{2} \)
41 \( 1 + 5iT - 41T^{2} \)
43 \( 1 + 6.71iT - 43T^{2} \)
47 \( 1 - 4.21iT - 47T^{2} \)
53 \( 1 - 5.41iT - 53T^{2} \)
59 \( 1 + 0.200iT - 59T^{2} \)
61 \( 1 + 6.21T + 61T^{2} \)
67 \( 1 + 3.65iT - 67T^{2} \)
71 \( 1 - 2.16T + 71T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 - 8.59iT - 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + 5.11T + 89T^{2} \)
97 \( 1 - 18.1T + 97T^{2} \)
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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

−11.34697847857735879986272994137, −10.66210265508309489909384810315, −8.848929639911043161841224547669, −7.919536102383048101194080501584, −6.95809355998763822053455960771, −6.08571834858261593727401199486, −5.10688421466135562494045695254, −4.27952379675166143968639157123, −3.06622797432229649902727759816, −1.89986457315361315310064572148, 2.14028122094074734697139140559, 3.61621874180764782650524549935, 4.25320144571447866704534169806, 5.07893293719504670311056956408, 5.99484503836233698460697258937, 7.25352530998413028390386314803, 7.942589265666442153705161350296, 9.412723838119498697964109535135, 10.63141805895315421492440369273, 11.43694907905260533219862984444

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