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Properties

Label 2-483-161.25-c1-0-4
Degree $2$
Conductor $483$
Sign $-0.829 + 0.559i$
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  + (−0.108 + 2.26i)2-s + (0.928 + 0.371i)3-s + (−3.14 − 0.300i)4-s + (0.756 + 3.11i)5-s + (−0.943 + 2.06i)6-s + (−2.46 − 0.954i)7-s + (0.373 − 2.60i)8-s + (0.723 + 0.690i)9-s + (−7.15 + 1.37i)10-s + (0.0489 + 1.02i)11-s + (−2.80 − 1.44i)12-s + (−0.882 − 1.01i)13-s + (2.43 − 5.49i)14-s + (−0.456 + 3.17i)15-s + (−0.341 − 0.0658i)16-s + (1.07 + 1.50i)17-s + ⋯
L(s)  = 1  + (−0.0764 + 1.60i)2-s + (0.535 + 0.214i)3-s + (−1.57 − 0.150i)4-s + (0.338 + 1.39i)5-s + (−0.385 + 0.843i)6-s + (−0.932 − 0.360i)7-s + (0.132 − 0.919i)8-s + (0.241 + 0.230i)9-s + (−2.26 + 0.436i)10-s + (0.0147 + 0.309i)11-s + (−0.809 − 0.417i)12-s + (−0.244 − 0.282i)13-s + (0.650 − 1.46i)14-s + (−0.117 + 0.820i)15-s + (−0.0854 − 0.0164i)16-s + (0.259 + 0.364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 + 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.829 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.829 + 0.559i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.829 + 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.377385 - 1.23478i\)
\(L(\frac12)\) \(\approx\) \(0.377385 - 1.23478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.928 - 0.371i)T \)
7 \( 1 + (2.46 + 0.954i)T \)
23 \( 1 + (0.750 - 4.73i)T \)
good2 \( 1 + (0.108 - 2.26i)T + (-1.99 - 0.190i)T^{2} \)
5 \( 1 + (-0.756 - 3.11i)T + (-4.44 + 2.29i)T^{2} \)
11 \( 1 + (-0.0489 - 1.02i)T + (-10.9 + 1.04i)T^{2} \)
13 \( 1 + (0.882 + 1.01i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-1.07 - 1.50i)T + (-5.56 + 16.0i)T^{2} \)
19 \( 1 + (-1.50 + 2.11i)T + (-6.21 - 17.9i)T^{2} \)
29 \( 1 + (-1.20 + 2.63i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-5.55 - 4.36i)T + (7.30 + 30.1i)T^{2} \)
37 \( 1 + (1.57 + 1.50i)T + (1.76 + 36.9i)T^{2} \)
41 \( 1 + (9.25 + 2.71i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-0.528 - 3.67i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (-2.39 - 4.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.285 - 0.824i)T + (-41.6 + 32.7i)T^{2} \)
59 \( 1 + (-12.3 + 2.38i)T + (54.7 - 21.9i)T^{2} \)
61 \( 1 + (-1.38 + 0.553i)T + (44.1 - 42.0i)T^{2} \)
67 \( 1 + (-7.78 + 4.01i)T + (38.8 - 54.5i)T^{2} \)
71 \( 1 + (-9.87 - 6.34i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (4.79 + 0.458i)T + (71.6 + 13.8i)T^{2} \)
79 \( 1 + (0.772 - 2.23i)T + (-62.0 - 48.8i)T^{2} \)
83 \( 1 + (4.20 - 1.23i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (10.4 - 8.23i)T + (20.9 - 86.4i)T^{2} \)
97 \( 1 + (15.6 + 4.59i)T + (81.6 + 52.4i)T^{2} \)
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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

−11.30514860520385486064688568458, −10.08757045481843481574823026590, −9.774718772024180470499491273650, −8.601016212814262074915897118275, −7.57484640286313381328506588988, −6.93791891561923523372641941017, −6.31594098034879956420893465556, −5.23032725330364923848829133413, −3.78219041044028094661960995302, −2.69733220166760507382929450336, 0.76502191860121772279404766616, 2.08976715516073533744587258536, 3.17787924165471040302883501684, 4.27198988021315240278629041075, 5.38397459982135949236664904213, 6.73651587286984267766938302437, 8.393213227039727463048085006616, 8.859235164167217882503457318833, 9.773949415129048621222186489342, 10.14581389141297683259059787865

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