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Properties

Label 2-483-161.9-c1-0-6
Degree $2$
Conductor $483$
Sign $-0.0103 - 0.999i$
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  + (−0.0723 + 0.298i)2-s + (−0.327 + 0.945i)3-s + (1.69 + 0.873i)4-s + (−2.45 − 0.983i)5-s + (−0.258 − 0.165i)6-s + (2.62 − 0.351i)7-s + (−0.785 + 0.906i)8-s + (−0.786 − 0.618i)9-s + (0.471 − 0.661i)10-s + (0.477 + 1.97i)11-s + (−1.37 + 1.31i)12-s + (1.94 + 4.26i)13-s + (−0.0850 + 0.807i)14-s + (1.73 − 2.00i)15-s + (1.99 + 2.80i)16-s + (0.115 − 2.43i)17-s + ⋯
L(s)  = 1  + (−0.0511 + 0.210i)2-s + (−0.188 + 0.545i)3-s + (0.846 + 0.436i)4-s + (−1.09 − 0.439i)5-s + (−0.105 − 0.0677i)6-s + (0.991 − 0.132i)7-s + (−0.277 + 0.320i)8-s + (−0.262 − 0.206i)9-s + (0.149 − 0.209i)10-s + (0.144 + 0.594i)11-s + (−0.398 + 0.379i)12-s + (0.540 + 1.18i)13-s + (−0.0227 + 0.215i)14-s + (0.447 − 0.516i)15-s + (0.499 + 0.701i)16-s + (0.0281 − 0.590i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0103 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0103 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.0103 - 0.999i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.0103 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.978134 + 0.988294i\)
\(L(\frac12)\) \(\approx\) \(0.978134 + 0.988294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.327 - 0.945i)T \)
7 \( 1 + (-2.62 + 0.351i)T \)
23 \( 1 + (0.778 - 4.73i)T \)
good2 \( 1 + (0.0723 - 0.298i)T + (-1.77 - 0.916i)T^{2} \)
5 \( 1 + (2.45 + 0.983i)T + (3.61 + 3.45i)T^{2} \)
11 \( 1 + (-0.477 - 1.97i)T + (-9.77 + 5.04i)T^{2} \)
13 \( 1 + (-1.94 - 4.26i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-0.115 + 2.43i)T + (-16.9 - 1.61i)T^{2} \)
19 \( 1 + (-0.130 - 2.73i)T + (-18.9 + 1.80i)T^{2} \)
29 \( 1 + (-6.42 - 4.12i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (5.74 + 1.10i)T + (28.7 + 11.5i)T^{2} \)
37 \( 1 + (6.70 + 5.27i)T + (8.72 + 35.9i)T^{2} \)
41 \( 1 + (0.609 + 4.24i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-2.70 - 3.12i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + (3.20 - 5.55i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-13.3 + 1.27i)T + (52.0 - 10.0i)T^{2} \)
59 \( 1 + (0.152 - 0.214i)T + (-19.2 - 55.7i)T^{2} \)
61 \( 1 + (4.13 + 11.9i)T + (-47.9 + 37.7i)T^{2} \)
67 \( 1 + (-4.56 - 4.35i)T + (3.18 + 66.9i)T^{2} \)
71 \( 1 + (-8.92 + 2.61i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (6.83 + 3.52i)T + (42.3 + 59.4i)T^{2} \)
79 \( 1 + (-6.29 - 0.601i)T + (77.5 + 14.9i)T^{2} \)
83 \( 1 + (-0.656 + 4.56i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-5.28 + 1.01i)T + (82.6 - 33.0i)T^{2} \)
97 \( 1 + (0.122 + 0.850i)T + (-93.0 + 27.3i)T^{2} \)
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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

−11.33617180418727053896872279483, −10.61547878755648863463643632301, −9.239722802715052530477381964342, −8.397403680775051963793148315814, −7.58318681814699381576143923352, −6.84601819188729653033883584520, −5.45687200553806614100628361142, −4.36406929707313559717977394643, −3.59128002175332594715799063945, −1.78184026404682485138235904512, 0.938958022563193127163473060650, 2.49728558023331368526756506694, 3.67236271539943378122650583034, 5.20230195344577258510296993525, 6.21398767739233664482007215963, 7.12632945318682055988663114262, 8.013664252713028979463143857108, 8.606748058257537477429898641518, 10.47046134405654074817666032383, 10.76099605746664067234174430279

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