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Properties

Label 2-483-161.61-c1-0-6
Degree $2$
Conductor $483$
Sign $0.636 - 0.771i$
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  + (−0.159 − 0.125i)2-s + (0.814 + 0.580i)3-s + (−0.461 − 1.90i)4-s + (−3.31 + 0.639i)5-s + (−0.0570 − 0.194i)6-s + (1.87 + 1.86i)7-s + (−0.333 + 0.729i)8-s + (0.327 + 0.945i)9-s + (0.608 + 0.313i)10-s + (3.22 + 4.09i)11-s + (0.728 − 1.81i)12-s + (−1.15 − 1.79i)13-s + (−0.0662 − 0.531i)14-s + (−3.07 − 1.40i)15-s + (−3.33 + 1.72i)16-s + (3.59 + 3.42i)17-s + ⋯
L(s)  = 1  + (−0.112 − 0.0885i)2-s + (0.470 + 0.334i)3-s + (−0.230 − 0.951i)4-s + (−1.48 + 0.286i)5-s + (−0.0232 − 0.0793i)6-s + (0.710 + 0.703i)7-s + (−0.117 + 0.257i)8-s + (0.109 + 0.315i)9-s + (0.192 + 0.0992i)10-s + (0.972 + 1.23i)11-s + (0.210 − 0.524i)12-s + (−0.319 − 0.497i)13-s + (−0.0176 − 0.142i)14-s + (−0.794 − 0.362i)15-s + (−0.834 + 0.430i)16-s + (0.871 + 0.830i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.636 - 0.771i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.636 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08871 + 0.513145i\)
\(L(\frac12)\) \(\approx\) \(1.08871 + 0.513145i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.814 - 0.580i)T \)
7 \( 1 + (-1.87 - 1.86i)T \)
23 \( 1 + (-0.195 - 4.79i)T \)
good2 \( 1 + (0.159 + 0.125i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (3.31 - 0.639i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (-3.22 - 4.09i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (1.15 + 1.79i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-3.59 - 3.42i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (-2.34 + 2.23i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (-5.28 + 1.55i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-0.552 - 5.78i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (-9.27 + 3.21i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (3.67 - 3.18i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (10.7 - 4.92i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (9.15 + 5.28i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.978 - 0.0466i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (2.02 - 3.93i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (4.91 + 6.89i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (0.244 + 0.611i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (-1.59 + 11.0i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-7.21 + 1.75i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (-10.6 + 0.508i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (7.36 - 8.49i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (8.61 + 0.822i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (2.77 + 3.20i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.19287064520143475077355564653, −10.08564923013984095088537458143, −9.439323117524387237295652687441, −8.363294967997029669729665566503, −7.71821073803068899481271978942, −6.58337843542008696830466269932, −5.13692865593149313319030871711, −4.44312289017259184497169378983, −3.22360205359666127634965752350, −1.56751374316146753325409698197, 0.823107412714376704578833985918, 3.12996286559581875751188878570, 3.90400399297596372564986280628, 4.72414003031492722014296204760, 6.64508137289342960285346939585, 7.51084623159454933755243772570, 8.164173239077551346802592122978, 8.607158458291508568126457090607, 9.762348321284801497419819746465, 11.33125820665769840714376411804

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