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Properties

Label 2-483-7.2-c1-0-8
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  + (1.13 − 1.96i)2-s + (0.5 + 0.866i)3-s + (−1.58 − 2.74i)4-s + (−2.07 + 3.59i)5-s + 2.27·6-s + (−1.69 + 2.03i)7-s − 2.64·8-s + (−0.499 + 0.866i)9-s + (4.71 + 8.17i)10-s + (1.59 + 2.76i)11-s + (1.58 − 2.74i)12-s + 0.175·13-s + (2.08 + 5.64i)14-s − 4.15·15-s + (0.156 − 0.270i)16-s + (0.663 + 1.14i)17-s + ⋯
L(s)  = 1  + (0.803 − 1.39i)2-s + (0.288 + 0.499i)3-s + (−0.791 − 1.37i)4-s + (−0.928 + 1.60i)5-s + 0.927·6-s + (−0.639 + 0.768i)7-s − 0.936·8-s + (−0.166 + 0.288i)9-s + (1.49 + 2.58i)10-s + (0.480 + 0.832i)11-s + (0.456 − 0.791i)12-s + 0.0487·13-s + (0.556 + 1.50i)14-s − 1.07·15-s + (0.0390 − 0.0676i)16-s + (0.160 + 0.278i)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

\(L(1)\) \(\approx\) \(1.67621 + 0.354493i\)
\(L(\frac12)\) \(\approx\) \(1.67621 + 0.354493i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.13 + 1.96i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (2.07 - 3.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.59 - 2.76i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.175T + 13T^{2} \)
17 \( 1 + (-0.663 - 1.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0252 - 0.0436i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 1.10T + 29T^{2} \)
31 \( 1 + (-4.35 - 7.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.88 - 3.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.84T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + (-4.47 + 7.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.98 - 8.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.19 - 5.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.61 + 6.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.01 + 5.21i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.71T + 71T^{2} \)
73 \( 1 + (-2.30 - 3.99i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.74 + 13.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.21T + 83T^{2} \)
89 \( 1 + (-0.970 + 1.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−11.09630965071401166465835123607, −10.32231176298318989741328681899, −9.840737815364508812920517492763, −8.586262837046732801763725509156, −7.25862011868388112799848448388, −6.31288381194569976460868929205, −4.86361504532602463103856251409, −3.77459631805805611650195193353, −3.16416138126449895106517805066, −2.27501025536518949375066765892, 0.814090694750727083414636215371, 3.58743315325005279377968649464, 4.25695546895641842709474458823, 5.31697640186767448771558463839, 6.30683705125668241752220823999, 7.26819887392800533794004781951, 8.028603684231789253958443974596, 8.627259998433891908405580413631, 9.609712346021600688375234829901, 11.29123271838147190855394576787

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