Properties

Label 2-483-161.32-c1-0-23
Degree $2$
Conductor $483$
Sign $-0.262 + 0.964i$
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  + (−1.09 − 1.04i)2-s + (0.981 − 0.189i)3-s + (0.0138 + 0.289i)4-s + (2.20 − 1.73i)5-s + (−1.27 − 0.818i)6-s + (2.57 − 0.626i)7-s + (−1.69 + 1.95i)8-s + (0.928 − 0.371i)9-s + (−4.22 − 0.403i)10-s + (1.54 − 1.47i)11-s + (0.0684 + 0.281i)12-s + (−0.526 − 1.15i)13-s + (−3.46 − 1.99i)14-s + (1.83 − 2.11i)15-s + (4.47 − 0.427i)16-s + (−1.62 + 0.839i)17-s + ⋯
L(s)  = 1  + (−0.774 − 0.738i)2-s + (0.566 − 0.109i)3-s + (0.00690 + 0.144i)4-s + (0.985 − 0.774i)5-s + (−0.519 − 0.334i)6-s + (0.971 − 0.236i)7-s + (−0.599 + 0.691i)8-s + (0.309 − 0.123i)9-s + (−1.33 − 0.127i)10-s + (0.465 − 0.443i)11-s + (0.0197 + 0.0814i)12-s + (−0.145 − 0.319i)13-s + (−0.927 − 0.534i)14-s + (0.473 − 0.546i)15-s + (1.11 − 0.106i)16-s + (−0.394 + 0.203i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.262 + 0.964i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.262 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886844 - 1.16010i\)
\(L(\frac12)\) \(\approx\) \(0.886844 - 1.16010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.981 + 0.189i)T \)
7 \( 1 + (-2.57 + 0.626i)T \)
23 \( 1 + (1.94 - 4.38i)T \)
good2 \( 1 + (1.09 + 1.04i)T + (0.0951 + 1.99i)T^{2} \)
5 \( 1 + (-2.20 + 1.73i)T + (1.17 - 4.85i)T^{2} \)
11 \( 1 + (-1.54 + 1.47i)T + (0.523 - 10.9i)T^{2} \)
13 \( 1 + (0.526 + 1.15i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (1.62 - 0.839i)T + (9.86 - 13.8i)T^{2} \)
19 \( 1 + (-6.07 - 3.13i)T + (11.0 + 15.4i)T^{2} \)
29 \( 1 + (6.31 + 4.05i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-1.62 - 4.68i)T + (-24.3 + 19.1i)T^{2} \)
37 \( 1 + (7.35 - 2.94i)T + (26.7 - 25.5i)T^{2} \)
41 \( 1 + (0.570 + 3.96i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (2.40 + 2.77i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + (5.93 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.98 - 8.40i)T + (-17.3 + 50.0i)T^{2} \)
59 \( 1 + (9.17 + 0.876i)T + (57.9 + 11.1i)T^{2} \)
61 \( 1 + (-0.974 - 0.187i)T + (56.6 + 22.6i)T^{2} \)
67 \( 1 + (0.561 - 2.31i)T + (-59.5 - 30.7i)T^{2} \)
71 \( 1 + (11.1 - 3.27i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-0.133 - 2.81i)T + (-72.6 + 6.93i)T^{2} \)
79 \( 1 + (-0.220 + 0.308i)T + (-25.8 - 74.6i)T^{2} \)
83 \( 1 + (-0.292 + 2.03i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (2.31 - 6.69i)T + (-69.9 - 55.0i)T^{2} \)
97 \( 1 + (-0.884 - 6.15i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.45284512691224723909627116467, −9.809405186314293237465406678499, −9.011561506043261582930560417976, −8.426564516579134533414831056050, −7.43277455532621961353179680824, −5.80661278146481154128504108282, −5.13936342113715285231961832560, −3.51802362541233690328354177415, −1.94696576082881207031318753433, −1.28150969946954600015720214027, 1.86365831276532809307566714300, 3.11180202637814957862167404226, 4.63985123875274469411762714101, 5.95210513087935650346483636004, 6.94653260893427752932817738969, 7.56303397292232242269929614336, 8.608252901111135118525011445186, 9.358610209506063662790349848704, 9.928209982760229471411017386150, 11.05466531330258887549801379231

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