Properties

Label 2-483-23.4-c1-0-5
Degree $2$
Conductor $483$
Sign $-0.304 - 0.952i$
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  + (0.698 + 1.53i)2-s + (−0.959 + 0.281i)3-s + (−0.544 + 0.627i)4-s + (2.35 + 1.51i)5-s + (−1.10 − 1.27i)6-s + (−0.142 − 0.989i)7-s + (1.88 + 0.554i)8-s + (0.841 − 0.540i)9-s + (−0.671 + 4.67i)10-s + (−1.24 + 2.72i)11-s + (0.345 − 0.755i)12-s + (0.518 − 3.60i)13-s + (1.41 − 0.909i)14-s + (−2.69 − 0.790i)15-s + (0.707 + 4.92i)16-s + (4.22 + 4.87i)17-s + ⋯
L(s)  = 1  + (0.494 + 1.08i)2-s + (−0.553 + 0.162i)3-s + (−0.272 + 0.313i)4-s + (1.05 + 0.678i)5-s + (−0.449 − 0.519i)6-s + (−0.0537 − 0.374i)7-s + (0.667 + 0.195i)8-s + (0.280 − 0.180i)9-s + (−0.212 + 1.47i)10-s + (−0.375 + 0.821i)11-s + (0.0996 − 0.218i)12-s + (0.143 − 0.999i)13-s + (0.378 − 0.243i)14-s + (−0.694 − 0.204i)15-s + (0.176 + 1.23i)16-s + (1.02 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.304 - 0.952i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.304 - 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15199 + 1.57744i\)
\(L(\frac12)\) \(\approx\) \(1.15199 + 1.57744i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (4.70 - 0.912i)T \)
good2 \( 1 + (-0.698 - 1.53i)T + (-1.30 + 1.51i)T^{2} \)
5 \( 1 + (-2.35 - 1.51i)T + (2.07 + 4.54i)T^{2} \)
11 \( 1 + (1.24 - 2.72i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.518 + 3.60i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-4.22 - 4.87i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-1.48 + 1.70i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (0.423 + 0.488i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (3.99 + 1.17i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (0.667 - 0.428i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (4.59 + 2.95i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (5.97 - 1.75i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 - 3.07T + 47T^{2} \)
53 \( 1 + (0.101 + 0.707i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-1.48 + 10.3i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-8.82 - 2.59i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-1.19 - 2.60i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (0.858 + 1.87i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-8.75 + 10.1i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-0.361 + 2.51i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-13.8 + 8.88i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-2.22 + 0.653i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (11.9 + 7.69i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.97034099924300583596115068023, −10.24508719362467318515735072194, −9.861678578021317770176390492932, −8.139379180632257104434940723199, −7.33634217909688960611685844325, −6.42580055943255466039105089188, −5.74032432804135449953509532752, −5.05194210565391499299920917625, −3.65685795548784545791508685750, −1.87512439456795571361618452764, 1.26867274057792952554276991963, 2.37833525596980750603266116783, 3.71079999384690796647124832231, 5.08709180932433246102961888808, 5.61841354907748316943449384796, 6.84259877512156687755284031652, 8.100454095635288589073209491715, 9.341085634484492626865074909388, 9.956274189902623858626268629367, 10.89797481249648598738509999191

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