Properties

Label 2-483-7.2-c1-0-17
Degree $2$
Conductor $483$
Sign $0.914 - 0.404i$
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  + (0.296 − 0.514i)2-s + (0.5 + 0.866i)3-s + (0.823 + 1.42i)4-s + (1.57 − 2.72i)5-s + 0.593·6-s + (−1.69 + 2.03i)7-s + 2.16·8-s + (−0.499 + 0.866i)9-s + (−0.935 − 1.61i)10-s + (3.27 + 5.66i)11-s + (−0.823 + 1.42i)12-s − 0.692·13-s + (0.543 + 1.47i)14-s + 3.15·15-s + (−1.00 + 1.74i)16-s + (−3.42 − 5.92i)17-s + ⋯
L(s)  = 1  + (0.209 − 0.363i)2-s + (0.288 + 0.499i)3-s + (0.411 + 0.713i)4-s + (0.704 − 1.22i)5-s + 0.242·6-s + (−0.639 + 0.768i)7-s + 0.765·8-s + (−0.166 + 0.288i)9-s + (−0.295 − 0.512i)10-s + (0.986 + 1.70i)11-s + (−0.237 + 0.411i)12-s − 0.192·13-s + (0.145 + 0.393i)14-s + 0.813·15-s + (−0.251 + 0.435i)16-s + (−0.829 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.914 - 0.404i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.914 - 0.404i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04229 + 0.431913i\)
\(L(\frac12)\) \(\approx\) \(2.04229 + 0.431913i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 2.03i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.296 + 0.514i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.57 + 2.72i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.27 - 5.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.692T + 13T^{2} \)
17 \( 1 + (3.42 + 5.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.45 + 2.52i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 2.94T + 29T^{2} \)
31 \( 1 + (-0.269 - 0.467i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.44 + 2.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.26T + 41T^{2} \)
43 \( 1 + 4.27T + 43T^{2} \)
47 \( 1 + (5.23 - 9.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.28 + 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.35 + 5.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.65 - 4.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.36 - 2.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.587T + 71T^{2} \)
73 \( 1 + (5.26 + 9.11i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.69 + 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.77T + 83T^{2} \)
89 \( 1 + (1.46 - 2.53i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.14T + 97T^{2} \)
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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

−11.26399091036334282565366346366, −9.749796047801452379693986412292, −9.411183495787148157023412211845, −8.692616098581943303882929745226, −7.37810412619521469930629922477, −6.44709475697830958461055744220, −4.92482152381655216821917453318, −4.44299791313645296770000703258, −2.89414889421412008751985900622, −1.90468433398070680125357429628, 1.36912929656625307713918935196, 2.83401178898302317099528592050, 3.91852542034319857725325483857, 5.86214817924593817550864922138, 6.35984901909854481435192542311, 6.84144933228519358112481450460, 8.000731223951189169410719715826, 9.237550612870776617443420711505, 10.23133908030084660735366680490, 10.79827734946430233987386098200

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