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Properties

Label 2-483-7.4-c1-0-8
Degree $2$
Conductor $483$
Sign $-0.00566 - 0.999i$
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  + (0.916 + 1.58i)2-s + (0.5 − 0.866i)3-s + (−0.679 + 1.17i)4-s + (0.471 + 0.817i)5-s + 1.83·6-s + (−1.16 + 2.37i)7-s + 1.17·8-s + (−0.499 − 0.866i)9-s + (−0.864 + 1.49i)10-s + (−1.04 + 1.80i)11-s + (0.679 + 1.17i)12-s + 3.64·13-s + (−4.83 + 0.334i)14-s + 0.943·15-s + (2.43 + 4.21i)16-s + (−0.149 + 0.259i)17-s + ⋯
L(s)  = 1  + (0.647 + 1.12i)2-s + (0.288 − 0.499i)3-s + (−0.339 + 0.588i)4-s + (0.210 + 0.365i)5-s + 0.748·6-s + (−0.439 + 0.898i)7-s + 0.415·8-s + (−0.166 − 0.288i)9-s + (−0.273 + 0.473i)10-s + (−0.314 + 0.544i)11-s + (0.196 + 0.339i)12-s + 1.01·13-s + (−1.29 + 0.0893i)14-s + 0.243·15-s + (0.608 + 1.05i)16-s + (−0.0363 + 0.0628i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00566 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00566 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.00566 - 0.999i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.00566 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60087 + 1.60996i\)
\(L(\frac12)\) \(\approx\) \(1.60087 + 1.60996i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.16 - 2.37i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.916 - 1.58i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.471 - 0.817i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.04 - 1.80i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.64T + 13T^{2} \)
17 \( 1 + (0.149 - 0.259i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.289 - 0.501i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 5.89T + 29T^{2} \)
31 \( 1 + (-3.01 + 5.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.618 - 1.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.42T + 41T^{2} \)
43 \( 1 + 9.08T + 43T^{2} \)
47 \( 1 + (2.53 + 4.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.16 - 2.01i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.04 - 1.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.22 + 10.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.06 + 3.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + (-5.78 + 10.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.57 + 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + (1.33 + 2.31i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.589T + 97T^{2} \)
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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

−11.31943129519473864809964093554, −10.25039167203463404845475595878, −9.183885777598567976236292474813, −8.202758914649987690220351051500, −7.40248036368731849713013660595, −6.33966484907019640801860735630, −5.97985486796257054031327607732, −4.78706547226465535112009019700, −3.40820714743758520377596905568, −2.00571890132037533528108215260, 1.28530776677899387597840307719, 2.94169066048299724731576268157, 3.71138163520756737258412725032, 4.62187852018049999662557419581, 5.71852072363880972102395892438, 7.10062991214134717225019955513, 8.229161786722884089575418641586, 9.259334112362935583288197346154, 10.15432215412178354756964325745, 10.89668122460524260619419260370

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