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Properties

Label 2-462-77.10-c1-0-8
Degree $2$
Conductor $462$
Sign $0.481 + 0.876i$
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  + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (1.54 − 0.893i)5-s − 0.999·6-s + (0.165 + 2.64i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.893 − 1.54i)10-s + (0.692 − 3.24i)11-s + (−0.866 + 0.499i)12-s + 6.37·13-s + (1.46 + 2.20i)14-s − 1.78·15-s + (−0.5 − 0.866i)16-s + (0.0530 − 0.0918i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.692 − 0.399i)5-s − 0.408·6-s + (0.0625 + 0.998i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.282 − 0.489i)10-s + (0.208 − 0.977i)11-s + (−0.249 + 0.144i)12-s + 1.76·13-s + (0.391 + 0.589i)14-s − 0.461·15-s + (−0.125 − 0.216i)16-s + (0.0128 − 0.0222i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.481 + 0.876i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.481 + 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69689 - 1.00363i\)
\(L(\frac12)\) \(\approx\) \(1.69689 - 1.00363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.165 - 2.64i)T \)
11 \( 1 + (-0.692 + 3.24i)T \)
good5 \( 1 + (-1.54 + 0.893i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 6.37T + 13T^{2} \)
17 \( 1 + (-0.0530 + 0.0918i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.07 + 3.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.97 + 6.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.65iT - 29T^{2} \)
31 \( 1 + (-1.10 - 0.639i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.26 + 3.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.0321T + 41T^{2} \)
43 \( 1 - 6.87iT - 43T^{2} \)
47 \( 1 + (-8.55 + 4.94i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.313 + 0.543i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.7 - 6.22i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.97 - 8.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.43 - 9.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.42T + 71T^{2} \)
73 \( 1 + (-0.0625 + 0.108i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.80 - 5.08i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + (4.64 - 2.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.7iT - 97T^{2} \)
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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

−11.08599271533791829507733778936, −10.31187468331921881697486318948, −8.863582758290988284953755677811, −8.590842802810633801218606429122, −6.75422058751526350344639925431, −5.90821535789960382731785657971, −5.46191394408148323902472829383, −4.08504376547770225403776570670, −2.65730597214918702050135405770, −1.29680997988908402248701741482, 1.75050404877217984461832570197, 3.67533186547858457783088076334, 4.31713705482390381448238015818, 5.75959056996073590163251927507, 6.29996837953928125996778098687, 7.29299833226499102054093345970, 8.306355327991709056161962093169, 9.741221386351934057353167373321, 10.31042359683760066176814957603, 11.21538846903723503321690140536

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