Properties

Label 2-483-7.4-c1-0-12
Degree $2$
Conductor $483$
Sign $0.434 + 0.900i$
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  + (−0.432 − 0.748i)2-s + (0.5 − 0.866i)3-s + (0.626 − 1.08i)4-s + (1.24 + 2.16i)5-s − 0.864·6-s + (−0.0271 + 2.64i)7-s − 2.81·8-s + (−0.499 − 0.866i)9-s + (1.07 − 1.86i)10-s + (1.99 − 3.44i)11-s + (−0.626 − 1.08i)12-s + 2.80·13-s + (1.99 − 1.12i)14-s + 2.49·15-s + (−0.0357 − 0.0619i)16-s + (3.49 − 6.05i)17-s + ⋯
L(s)  = 1  + (−0.305 − 0.529i)2-s + (0.288 − 0.499i)3-s + (0.313 − 0.542i)4-s + (0.558 + 0.966i)5-s − 0.353·6-s + (−0.0102 + 0.999i)7-s − 0.994·8-s + (−0.166 − 0.288i)9-s + (0.341 − 0.591i)10-s + (0.600 − 1.04i)11-s + (−0.180 − 0.313i)12-s + 0.778·13-s + (0.532 − 0.300i)14-s + 0.644·15-s + (−0.00894 − 0.0154i)16-s + (0.847 − 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.434 + 0.900i$
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: \((2,\ 483,\ (\ :1/2),\ 0.434 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37474 - 0.862691i\)
\(L(\frac12)\) \(\approx\) \(1.37474 - 0.862691i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0271 - 2.64i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.432 + 0.748i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.24 - 2.16i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.99 + 3.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.80T + 13T^{2} \)
17 \( 1 + (-3.49 + 6.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.99 - 5.17i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 4.06T + 29T^{2} \)
31 \( 1 + (-1.72 + 2.98i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.738 - 1.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.65T + 41T^{2} \)
43 \( 1 - 7.07T + 43T^{2} \)
47 \( 1 + (-1.52 - 2.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.78 - 8.28i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.15 + 7.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.85 - 3.21i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.23 - 3.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.49T + 71T^{2} \)
73 \( 1 + (6.45 - 11.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.56 + 6.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.05T + 83T^{2} \)
89 \( 1 + (-1.65 - 2.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−10.90532052544071764577806334947, −9.867451638251130557867174900420, −9.268710727781851064939757134387, −8.291075668109997725790283912775, −7.03925687069943382750532849450, −5.98875920542086182822335793854, −5.70810394744731569329609296263, −3.31105075833061559200332888765, −2.61580410959415691953475459909, −1.31212976995492606460724433925, 1.55169961043904939226911510325, 3.41861610153160622726538345030, 4.33194204683434265306398489701, 5.54958863202616760237125744349, 6.70323495458894264875215061939, 7.57589379409704157671502402070, 8.502895910404942852417957893547, 9.229034944857441985155165092349, 10.00158084671690728336247376733, 11.02432667552776657784788947025

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