Properties

Label 2-483-161.18-c1-0-19
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: $\chi_{483} (340, \cdot )$
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

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

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