Properties

Label 2-462-33.29-c1-0-21
Degree $2$
Conductor $462$
Sign $-0.990 + 0.140i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
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  + (−0.809 − 0.587i)2-s + (0.455 + 1.67i)3-s + (0.309 + 0.951i)4-s + (−1.96 − 2.70i)5-s + (0.613 − 1.61i)6-s + (0.951 − 0.309i)7-s + (0.309 − 0.951i)8-s + (−2.58 + 1.52i)9-s + 3.34i·10-s + (−3.09 − 1.19i)11-s + (−1.44 + 0.949i)12-s + (−1.74 + 2.40i)13-s + (−0.951 − 0.309i)14-s + (3.63 − 4.52i)15-s + (−0.809 + 0.587i)16-s + (−1.58 + 1.15i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.263 + 0.964i)3-s + (0.154 + 0.475i)4-s + (−0.880 − 1.21i)5-s + (0.250 − 0.661i)6-s + (0.359 − 0.116i)7-s + (0.109 − 0.336i)8-s + (−0.861 + 0.507i)9-s + 1.05i·10-s + (−0.932 − 0.361i)11-s + (−0.418 + 0.274i)12-s + (−0.483 + 0.665i)13-s + (−0.254 − 0.0825i)14-s + (0.937 − 1.16i)15-s + (−0.202 + 0.146i)16-s + (−0.385 + 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.990 + 0.140i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.990 + 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00776646 - 0.110175i\)
\(L(\frac12)\) \(\approx\) \(0.00776646 - 0.110175i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-0.455 - 1.67i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (3.09 + 1.19i)T \)
good5 \( 1 + (1.96 + 2.70i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (1.74 - 2.40i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.58 - 1.15i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (5.05 + 1.64i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.29iT - 23T^{2} \)
29 \( 1 + (0.984 + 3.02i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.28 + 3.84i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.250 + 0.771i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.61 + 8.04i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.94iT - 43T^{2} \)
47 \( 1 + (4.03 + 1.31i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.18 - 9.88i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (6.46 - 2.10i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-9.05 - 12.4i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.291T + 67T^{2} \)
71 \( 1 + (-5.37 - 7.40i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-12.3 + 4.02i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.656 + 0.903i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.505 - 0.367i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 8.89iT - 89T^{2} \)
97 \( 1 + (7.11 + 5.16i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.72856995702570175975649971617, −9.587736574317274222566739296079, −8.826746423594847978468340726149, −8.259249518818419673149075533381, −7.40235624507734868767065472466, −5.54759840527251695963946592652, −4.51548482720559938440829038024, −3.85203784976090232793085588793, −2.24625443443109473802281020505, −0.07351508832188240891598392807, 2.16877612078297023192161460630, 3.21600905135447931854373799449, 4.97357181073722443723020154633, 6.37567804602617407079206038118, 7.01476004174103358723888831449, 7.948829375920054066197834992204, 8.216893553709624431453530802078, 9.636190672240448556040084254244, 10.82480909337553978163787767496, 11.14948310367312848617998159807

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