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Properties

Label 2-483-7.2-c1-0-9
Degree $2$
Conductor $483$
Sign $0.266 - 0.963i$
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  + (−0.5 − 0.866i)3-s + (1 + 1.73i)4-s + (0.5 + 2.59i)7-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)11-s + (0.999 − 1.73i)12-s − 3·13-s + (−1.99 + 3.46i)16-s + (−1 − 1.73i)17-s + (−1.5 + 2.59i)19-s + (2 − 1.73i)21-s + (−0.5 + 0.866i)23-s + (2.5 + 4.33i)25-s + 0.999·27-s + (−4 + 3.46i)28-s + 6·29-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.5 + 0.866i)4-s + (0.188 + 0.981i)7-s + (−0.166 + 0.288i)9-s + (0.301 + 0.522i)11-s + (0.288 − 0.499i)12-s − 0.832·13-s + (−0.499 + 0.866i)16-s + (−0.242 − 0.420i)17-s + (−0.344 + 0.596i)19-s + (0.436 − 0.377i)21-s + (−0.104 + 0.180i)23-s + (0.5 + 0.866i)25-s + 0.192·27-s + (−0.755 + 0.654i)28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.266 - 0.963i$
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: \((2,\ 483,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04351 + 0.793861i\)
\(L(\frac12)\) \(\approx\) \(1.04351 + 0.793861i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 3T + 43T^{2} \)
47 \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−11.47336983773104904235557683552, −10.42045810761469081055717817051, −9.198462570764373662748236291977, −8.389462246331985922386263755143, −7.42692095864344879048971388049, −6.74025870014379059487372318660, −5.63648180212877538251114042523, −4.47962722379806541495878870072, −2.94760024981985077568430727468, −1.96328165780901198634610387737, 0.830448254714121241522396090724, 2.59920772360186432263395704160, 4.20401699536831176714123535759, 4.99980390802012691916442982875, 6.23477990751254073746139050507, 6.86600108735784035500470346867, 8.068739417884246567022959148542, 9.256790894451603555770324442040, 10.16436477849446367153882541146, 10.68483642553712540580669288659

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