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Properties

Label 2-483-7.2-c1-0-29
Degree $2$
Conductor $483$
Sign $-0.615 - 0.788i$
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  + (1.27 − 2.21i)2-s + (−0.5 − 0.866i)3-s + (−2.25 − 3.90i)4-s + (−0.324 + 0.561i)5-s − 2.55·6-s + (−2.38 − 1.14i)7-s − 6.41·8-s + (−0.499 + 0.866i)9-s + (0.827 + 1.43i)10-s + (1.49 + 2.58i)11-s + (−2.25 + 3.90i)12-s − 3.87·13-s + (−5.57 + 3.80i)14-s + 0.648·15-s + (−3.67 + 6.36i)16-s + (−3.36 − 5.83i)17-s + ⋯
L(s)  = 1  + (0.902 − 1.56i)2-s + (−0.288 − 0.499i)3-s + (−1.12 − 1.95i)4-s + (−0.144 + 0.251i)5-s − 1.04·6-s + (−0.901 − 0.433i)7-s − 2.26·8-s + (−0.166 + 0.288i)9-s + (0.261 + 0.453i)10-s + (0.449 + 0.778i)11-s + (−0.651 + 1.12i)12-s − 1.07·13-s + (−1.49 + 1.01i)14-s + 0.167·15-s + (−0.918 + 1.59i)16-s + (−0.816 − 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.615 - 0.788i$
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: \((2,\ 483,\ (\ :1/2),\ -0.615 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.555264 + 1.13767i\)
\(L(\frac12)\) \(\approx\) \(0.555264 + 1.13767i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.38 + 1.14i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.27 + 2.21i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.324 - 0.561i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.49 - 2.58i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.87T + 13T^{2} \)
17 \( 1 + (3.36 + 5.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.48 + 6.04i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 4.40T + 29T^{2} \)
31 \( 1 + (1.10 + 1.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.20 + 3.81i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.194T + 41T^{2} \)
43 \( 1 + 1.37T + 43T^{2} \)
47 \( 1 + (-1.24 + 2.15i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.73 + 8.20i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.40 + 5.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.36 - 7.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.446 - 0.774i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.2T + 71T^{2} \)
73 \( 1 + (0.471 + 0.817i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.07 - 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + (-7.87 + 13.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.23T + 97T^{2} \)
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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

−10.74960635144348611430312667195, −9.635715542782995075944326022307, −9.329289271205424548760596338790, −7.25929229484002918410951137200, −6.71598362384242675536923022595, −5.18361068260413551153257335095, −4.50164325765179606199257752422, −3.14731536199204428234287778428, −2.33502641153273932517313702129, −0.58662281139350460347858763004, 3.19745798577592455100680524184, 4.13228386297394765128720579758, 5.11434456975839068527414538119, 6.10552252638998225729733142897, 6.51440442440706716972622606238, 7.81851213892711664419995244638, 8.618074099873188055734498972707, 9.490818885792182500591036628664, 10.59158311619615529487519658810, 12.14997182839883506028190038405

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