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Properties

Label 2-483-161.19-c1-0-9
Degree $2$
Conductor $483$
Sign $0.576 - 0.816i$
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  + (0.174 + 0.0166i)2-s + (−0.690 + 0.723i)3-s + (−1.93 − 0.372i)4-s + (0.315 − 0.162i)5-s + (−0.132 + 0.114i)6-s + (2.63 + 0.276i)7-s + (−0.666 − 0.195i)8-s + (−0.0475 − 0.998i)9-s + (0.0576 − 0.0230i)10-s + (0.132 + 1.38i)11-s + (1.60 − 1.14i)12-s + (−0.399 − 0.0574i)13-s + (0.453 + 0.0919i)14-s + (−0.100 + 0.340i)15-s + (3.54 + 1.41i)16-s + (0.00865 − 0.0250i)17-s + ⋯
L(s)  = 1  + (0.123 + 0.0117i)2-s + (−0.398 + 0.417i)3-s + (−0.966 − 0.186i)4-s + (0.141 − 0.0727i)5-s + (−0.0539 + 0.0467i)6-s + (0.994 + 0.104i)7-s + (−0.235 − 0.0691i)8-s + (−0.0158 − 0.332i)9-s + (0.0182 − 0.00730i)10-s + (0.0398 + 0.417i)11-s + (0.463 − 0.329i)12-s + (−0.110 − 0.0159i)13-s + (0.121 + 0.0245i)14-s + (−0.0258 + 0.0879i)15-s + (0.885 + 0.354i)16-s + (0.00209 − 0.00606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.576 - 0.816i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.576 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02017 + 0.528625i\)
\(L(\frac12)\) \(\approx\) \(1.02017 + 0.528625i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.690 - 0.723i)T \)
7 \( 1 + (-2.63 - 0.276i)T \)
23 \( 1 + (-4.33 - 2.06i)T \)
good2 \( 1 + (-0.174 - 0.0166i)T + (1.96 + 0.378i)T^{2} \)
5 \( 1 + (-0.315 + 0.162i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (-0.132 - 1.38i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (0.399 + 0.0574i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (-0.00865 + 0.0250i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (-1.86 - 5.39i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (-3.18 - 3.67i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-4.35 + 1.05i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (-8.03 + 0.382i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (6.16 - 9.59i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (0.194 + 0.660i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (0.862 + 0.498i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.03 + 10.2i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (2.21 + 5.53i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (-2.73 + 2.60i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (-3.39 - 2.41i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (5.73 + 12.5i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (1.45 - 7.54i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (-6.42 + 8.16i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (8.45 - 5.43i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-3.10 + 12.8i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (-10.0 - 6.45i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.14936919067312046437105906849, −10.04862406324137962319578040277, −9.527992951901438943468093103697, −8.469775594251447363420561631375, −7.67730810310086625650984900511, −6.21041439044256509473153027871, −5.17331581687991520552203208499, −4.66517508068471934840966202605, −3.45363167350155025878424688170, −1.40403390885564852562837585413, 0.858827621088085544003775552556, 2.70993106258771137026918041977, 4.33087522470062627318713995895, 5.01933473400907088972330441014, 6.06166165627952665946827247612, 7.27227131513178537487373108833, 8.203808553435417088154951859399, 8.901922699268405635254338073946, 9.984649534150190680391497350932, 10.97632749667301317401493527938

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