Properties

Label 2-483-161.34-c1-0-7
Degree $2$
Conductor $483$
Sign $-0.713 - 0.701i$
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  + (−1.85 + 1.19i)2-s + (0.989 + 0.142i)3-s + (1.19 − 2.61i)4-s + (−1.10 + 0.324i)5-s + (−2.00 + 0.916i)6-s + (1.07 + 2.41i)7-s + (0.274 + 1.90i)8-s + (0.959 + 0.281i)9-s + (1.66 − 1.92i)10-s + (−0.0983 + 0.152i)11-s + (1.55 − 2.41i)12-s + (3.11 + 2.70i)13-s + (−4.88 − 3.19i)14-s + (−1.14 + 0.164i)15-s + (0.978 + 1.12i)16-s + (−2.24 − 4.91i)17-s + ⋯
L(s)  = 1  + (−1.31 + 0.843i)2-s + (0.571 + 0.0821i)3-s + (0.596 − 1.30i)4-s + (−0.494 + 0.145i)5-s + (−0.819 + 0.374i)6-s + (0.407 + 0.913i)7-s + (0.0969 + 0.674i)8-s + (0.319 + 0.0939i)9-s + (0.527 − 0.608i)10-s + (−0.0296 + 0.0461i)11-s + (0.448 − 0.697i)12-s + (0.864 + 0.749i)13-s + (−1.30 − 0.854i)14-s + (−0.294 + 0.0423i)15-s + (0.244 + 0.282i)16-s + (−0.544 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.713 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.713 - 0.701i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.713 - 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.287182 + 0.701758i\)
\(L(\frac12)\) \(\approx\) \(0.287182 + 0.701758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (-1.07 - 2.41i)T \)
23 \( 1 + (-0.108 - 4.79i)T \)
good2 \( 1 + (1.85 - 1.19i)T + (0.830 - 1.81i)T^{2} \)
5 \( 1 + (1.10 - 0.324i)T + (4.20 - 2.70i)T^{2} \)
11 \( 1 + (0.0983 - 0.152i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (-3.11 - 2.70i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (2.24 + 4.91i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (0.214 - 0.469i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-3.47 - 7.60i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-3.48 + 0.500i)T + (29.7 - 8.73i)T^{2} \)
37 \( 1 + (0.139 - 0.473i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (-2.65 - 9.05i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (9.24 + 1.32i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + 3.91iT - 47T^{2} \)
53 \( 1 + (5.66 - 4.91i)T + (7.54 - 52.4i)T^{2} \)
59 \( 1 + (-3.29 - 2.85i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (0.757 + 5.26i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (1.30 + 2.03i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (7.91 - 5.08i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (5.88 + 2.68i)T + (47.8 + 55.1i)T^{2} \)
79 \( 1 + (-2.99 - 2.59i)T + (11.2 + 78.1i)T^{2} \)
83 \( 1 + (-8.62 - 2.53i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-2.58 + 18.0i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-6.91 + 2.02i)T + (81.6 - 52.4i)T^{2} \)
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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

−11.22498350022909008704418385646, −10.01950731713344025574765723709, −9.155820391072733806393879240676, −8.688174922874614159985866309282, −7.86571805252360419721510001863, −7.06092448886667563358975583068, −6.12255590444873764968667764317, −4.79759409587526073958312777984, −3.27184657091619915096361999364, −1.62580134628004418938629082819, 0.70085248838139110288034668534, 2.05942472203700997633768773745, 3.44270036293876870259369816346, 4.42934290770682009573117668957, 6.28499708725321515276971053841, 7.58201118767935316270728680982, 8.284214098607259496061370683434, 8.631317956977326875806934773478, 9.964821122125420124572120537837, 10.49054317446101399285677605401

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