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Properties

Label 2-483-161.18-c1-0-29
Degree $2$
Conductor $483$
Sign $-0.683 + 0.729i$
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  + (−0.282 − 1.16i)2-s + (0.327 + 0.945i)3-s + (0.504 − 0.259i)4-s + (−0.840 + 0.336i)5-s + (1.00 − 0.647i)6-s + (−1.53 − 2.15i)7-s + (−2.01 − 2.32i)8-s + (−0.786 + 0.618i)9-s + (0.628 + 0.882i)10-s + (1.45 − 5.99i)11-s + (0.410 + 0.391i)12-s + (−2.00 + 4.39i)13-s + (−2.07 + 2.39i)14-s + (−0.592 − 0.683i)15-s + (−1.47 + 2.07i)16-s + (−0.0414 − 0.870i)17-s + ⋯
L(s)  = 1  + (−0.199 − 0.822i)2-s + (0.188 + 0.545i)3-s + (0.252 − 0.129i)4-s + (−0.375 + 0.150i)5-s + (0.411 − 0.264i)6-s + (−0.581 − 0.813i)7-s + (−0.711 − 0.821i)8-s + (−0.262 + 0.206i)9-s + (0.198 + 0.279i)10-s + (0.438 − 1.80i)11-s + (0.118 + 0.112i)12-s + (−0.556 + 1.21i)13-s + (−0.553 + 0.640i)14-s + (−0.153 − 0.176i)15-s + (−0.368 + 0.518i)16-s + (−0.0100 − 0.211i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.683 + 0.729i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (340, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.683 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.430377 - 0.993254i\)
\(L(\frac12)\) \(\approx\) \(0.430377 - 0.993254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.327 - 0.945i)T \)
7 \( 1 + (1.53 + 2.15i)T \)
23 \( 1 + (1.13 + 4.66i)T \)
good2 \( 1 + (0.282 + 1.16i)T + (-1.77 + 0.916i)T^{2} \)
5 \( 1 + (0.840 - 0.336i)T + (3.61 - 3.45i)T^{2} \)
11 \( 1 + (-1.45 + 5.99i)T + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (2.00 - 4.39i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (0.0414 + 0.870i)T + (-16.9 + 1.61i)T^{2} \)
19 \( 1 + (-0.205 + 4.30i)T + (-18.9 - 1.80i)T^{2} \)
29 \( 1 + (-3.69 + 2.37i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-7.34 + 1.41i)T + (28.7 - 11.5i)T^{2} \)
37 \( 1 + (-7.84 + 6.16i)T + (8.72 - 35.9i)T^{2} \)
41 \( 1 + (0.752 - 5.23i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (4.36 - 5.03i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (5.11 + 8.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.59 + 0.248i)T + (52.0 + 10.0i)T^{2} \)
59 \( 1 + (0.844 + 1.18i)T + (-19.2 + 55.7i)T^{2} \)
61 \( 1 + (4.04 - 11.6i)T + (-47.9 - 37.7i)T^{2} \)
67 \( 1 + (0.738 - 0.703i)T + (3.18 - 66.9i)T^{2} \)
71 \( 1 + (-8.14 - 2.39i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-7.44 + 3.83i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (-12.9 + 1.23i)T + (77.5 - 14.9i)T^{2} \)
83 \( 1 + (-1.45 - 10.1i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-2.26 - 0.436i)T + (82.6 + 33.0i)T^{2} \)
97 \( 1 + (-0.730 + 5.07i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.79432521362078865383426827527, −9.807496848408234148108251454964, −9.246426670478660112069322568019, −8.145824457652904460746907007651, −6.79953770145296625536621881435, −6.19801903949112741797416135112, −4.46967359921688340482700840615, −3.53751980982069614229522639887, −2.62297603387276236327902315497, −0.65982379396495195656499463819, 2.07484391543254657744125262987, 3.24112183856667142163548674208, 4.90945580426807176750197789109, 6.05255297263588674028579640642, 6.76504427072810430830873919451, 7.81850034703734954879034942206, 8.140587281092138653025035771143, 9.431752111193011052146054365494, 10.12425790703558555601365488817, 11.69024619151705002888441441235

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