Properties

Label 2-483-161.61-c1-0-4
Degree $2$
Conductor $483$
Sign $0.969 + 0.244i$
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  + (−0.929 − 0.730i)2-s + (−0.814 − 0.580i)3-s + (−0.142 − 0.585i)4-s + (−3.37 + 0.650i)5-s + (0.333 + 1.13i)6-s + (2.36 + 1.17i)7-s + (−1.27 + 2.79i)8-s + (0.327 + 0.945i)9-s + (3.60 + 1.86i)10-s + (−2.39 − 3.04i)11-s + (−0.224 + 0.559i)12-s + (1.88 + 2.93i)13-s + (−1.34 − 2.82i)14-s + (3.12 + 1.42i)15-s + (2.16 − 1.11i)16-s + (1.19 + 1.14i)17-s + ⋯
L(s)  = 1  + (−0.656 − 0.516i)2-s + (−0.470 − 0.334i)3-s + (−0.0710 − 0.292i)4-s + (−1.50 + 0.290i)5-s + (0.135 + 0.463i)6-s + (0.895 + 0.444i)7-s + (−0.451 + 0.989i)8-s + (0.109 + 0.315i)9-s + (1.14 + 0.588i)10-s + (−0.720 − 0.916i)11-s + (−0.0646 + 0.161i)12-s + (0.522 + 0.813i)13-s + (−0.358 − 0.754i)14-s + (0.806 + 0.368i)15-s + (0.540 − 0.278i)16-s + (0.290 + 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.969 + 0.244i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.969 + 0.244i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.562964 - 0.0699111i\)
\(L(\frac12)\) \(\approx\) \(0.562964 - 0.0699111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.814 + 0.580i)T \)
7 \( 1 + (-2.36 - 1.17i)T \)
23 \( 1 + (-4.73 + 0.781i)T \)
good2 \( 1 + (0.929 + 0.730i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (3.37 - 0.650i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (2.39 + 3.04i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (-1.88 - 2.93i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-1.19 - 1.14i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (1.99 - 1.90i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (-4.78 + 1.40i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (0.359 + 3.76i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (-2.75 + 0.952i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (0.0864 - 0.0749i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (3.62 - 1.65i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-7.77 - 4.49i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.77 - 0.179i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (2.01 - 3.90i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (-6.69 - 9.39i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (3.13 + 7.83i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (2.25 - 15.6i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (0.107 - 0.0260i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (-6.52 + 0.310i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (1.35 - 1.56i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-13.3 - 1.27i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (-12.5 - 14.5i)T + (-13.8 + 96.0i)T^{2} \)
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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

−10.99272861654331624443098301666, −10.48734176101784710406534434673, −8.964199080543716806297768967319, −8.293696595255821807035831281493, −7.65774054076391342086111169230, −6.30321560401081566546127242416, −5.27705102279980533895820179770, −4.14077235619189731507449880533, −2.59488077414473819940160758273, −0.988385084609548904694128196553, 0.63297872673905837138009208627, 3.31116139570883067782970727815, 4.37815843435914536252759144913, 5.12242311624978472171565045438, 6.85366874620423112348197686298, 7.55107581308443671534211080155, 8.184348199818533443577188286463, 8.909643134482878421330917863460, 10.22649220830689967825753574425, 10.92462365927715626560464295986

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