Properties

Label 2-483-7.2-c1-0-10
Degree $2$
Conductor $483$
Sign $0.171 - 0.985i$
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  + (−1.25 + 2.17i)2-s + (−0.5 − 0.866i)3-s + (−2.16 − 3.75i)4-s + (−1.20 + 2.09i)5-s + 2.51·6-s + (−0.750 − 2.53i)7-s + 5.87·8-s + (−0.499 + 0.866i)9-s + (−3.04 − 5.27i)10-s + (−0.624 − 1.08i)11-s + (−2.16 + 3.75i)12-s + 2.06·13-s + (6.47 + 1.55i)14-s + 2.41·15-s + (−3.05 + 5.29i)16-s + (1.65 + 2.86i)17-s + ⋯
L(s)  = 1  + (−0.889 + 1.54i)2-s + (−0.288 − 0.499i)3-s + (−1.08 − 1.87i)4-s + (−0.540 + 0.936i)5-s + 1.02·6-s + (−0.283 − 0.958i)7-s + 2.07·8-s + (−0.166 + 0.288i)9-s + (−0.962 − 1.66i)10-s + (−0.188 − 0.325i)11-s + (−0.625 + 1.08i)12-s + 0.573·13-s + (1.73 + 0.415i)14-s + 0.624·15-s + (−0.764 + 1.32i)16-s + (0.401 + 0.695i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.171 - 0.985i$
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: \((2,\ 483,\ (\ :1/2),\ 0.171 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.482415 + 0.405616i\)
\(L(\frac12)\) \(\approx\) \(0.482415 + 0.405616i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.750 + 2.53i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.25 - 2.17i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.20 - 2.09i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.624 + 1.08i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.06T + 13T^{2} \)
17 \( 1 + (-1.65 - 2.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.80 + 3.13i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 4.27T + 29T^{2} \)
31 \( 1 + (-3.98 - 6.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.54 - 4.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.885T + 41T^{2} \)
43 \( 1 - 9.94T + 43T^{2} \)
47 \( 1 + (-2.84 + 4.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.47 - 2.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.01 + 1.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.69 + 6.39i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.14 + 7.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.27T + 71T^{2} \)
73 \( 1 + (-5.99 - 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.61 + 13.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 + (-8.13 + 14.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−10.63168209476771714609140703262, −10.44295805820709914121983011932, −9.083127654448653214273761836184, −8.127190647989725470661021902987, −7.45017959539621182182882059938, −6.73559957051523342964343299971, −6.18057608237121716944048325691, −4.88569606443051781532927054766, −3.34620611933430143468251450172, −0.868939131197608328184619078604, 0.848818120217520339361478317460, 2.48999626566907119186137429413, 3.66864568913108044809901338913, 4.67347978235948021168986801208, 5.85575584847405817216213152560, 7.70857584811147498858320835993, 8.541405446651733617417979868080, 9.225910810860966159197706313338, 9.853822878111832175440296132905, 10.78122437250713739134044770142

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