Properties

Label 2-483-161.10-c1-0-27
Degree $2$
Conductor $483$
Sign $-0.848 + 0.529i$
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  + (0.571 − 1.65i)2-s + (−0.458 − 0.888i)3-s + (−0.831 − 0.654i)4-s + (−0.0705 + 0.00673i)5-s + (−1.73 + 0.248i)6-s + (0.803 − 2.52i)7-s + (1.38 − 0.890i)8-s + (−0.580 + 0.814i)9-s + (−0.0292 + 0.120i)10-s + (4.41 − 1.52i)11-s + (−0.200 + 1.03i)12-s + (0.351 + 1.19i)13-s + (−3.70 − 2.76i)14-s + (0.0383 + 0.0596i)15-s + (−1.17 − 4.85i)16-s + (−3.02 − 1.21i)17-s + ⋯
L(s)  = 1  + (0.404 − 1.16i)2-s + (−0.264 − 0.513i)3-s + (−0.415 − 0.327i)4-s + (−0.0315 + 0.00301i)5-s + (−0.706 + 0.101i)6-s + (0.303 − 0.952i)7-s + (0.489 − 0.314i)8-s + (−0.193 + 0.271i)9-s + (−0.00923 + 0.0380i)10-s + (1.33 − 0.460i)11-s + (−0.0578 + 0.299i)12-s + (0.0974 + 0.332i)13-s + (−0.990 − 0.740i)14-s + (0.00989 + 0.0153i)15-s + (−0.294 − 1.21i)16-s + (−0.733 − 0.293i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.848 + 0.529i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.848 + 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.484545 - 1.69189i\)
\(L(\frac12)\) \(\approx\) \(0.484545 - 1.69189i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.458 + 0.888i)T \)
7 \( 1 + (-0.803 + 2.52i)T \)
23 \( 1 + (1.12 - 4.66i)T \)
good2 \( 1 + (-0.571 + 1.65i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (0.0705 - 0.00673i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-4.41 + 1.52i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.351 - 1.19i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (3.02 + 1.21i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (3.72 - 1.49i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.615 + 4.27i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (0.962 - 0.0458i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-2.76 - 1.96i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-9.24 - 4.22i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-5.80 + 9.02i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-1.47 + 0.853i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.16 - 8.56i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-6.83 - 1.65i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (12.6 + 6.50i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-1.66 - 8.65i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-3.54 - 4.09i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (5.93 - 7.54i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-1.78 + 1.86i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-6.10 - 13.3i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.172 - 3.63i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-1.10 + 2.42i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.04112816044776472075433054169, −10.01888942022050249931387960148, −9.056103190174310325453303246353, −7.75549090336605407836171427146, −6.95192953544069470818096333309, −5.91962949734025113838398880105, −4.28893139078226509606178066661, −3.83169037567596259671657691228, −2.20155620273128716284857303701, −1.06924501679857893137831013191, 2.12834115810296500606892837886, 4.07784002325731658289250170233, 4.78412555225201778143556513389, 5.98709193727767924694483582122, 6.38766780371378681894995849768, 7.56957602893398284624927503497, 8.656338101656323605729568820174, 9.238307578154037142938357743967, 10.56359834253353268437753273155, 11.31215974987699547496774866124

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