Properties

Label 2-462-21.5-c1-0-5
Degree $2$
Conductor $462$
Sign $-0.667 - 0.744i$
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 + 1.5i)3-s + (0.499 + 0.866i)4-s + (−1.73 + 3i)5-s − 1.73i·6-s + (2 + 1.73i)7-s − 0.999i·8-s + (−1.5 + 2.59i)9-s + (3 − 1.73i)10-s + (−0.866 + 0.5i)11-s + (−0.866 + 1.49i)12-s − 3.46i·13-s + (−0.866 − 2.5i)14-s − 6·15-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 + 0.866i)3-s + (0.249 + 0.433i)4-s + (−0.774 + 1.34i)5-s − 0.707i·6-s + (0.755 + 0.654i)7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (0.948 − 0.547i)10-s + (−0.261 + 0.150i)11-s + (−0.250 + 0.433i)12-s − 0.960i·13-s + (−0.231 − 0.668i)14-s − 1.54·15-s + (−0.125 + 0.216i)16-s + (0.210 + 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.667 - 0.744i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.667 - 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.393309 + 0.880167i\)
\(L(\frac12)\) \(\approx\) \(0.393309 + 0.880167i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T \)
7 \( 1 + (-2 - 1.73i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (1.73 - 3i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (-0.866 - 1.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-0.866 + 1.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.06 + 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9 - 5.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8 - 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15iT - 71T^{2} \)
73 \( 1 + (12 - 6.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.5iT - 97T^{2} \)
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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

−11.11156792867219420626629246180, −10.49459781212864453831899850981, −9.828203399056617367844100048552, −8.482270700801752333784414896482, −8.099809327555687157242263061280, −7.11811583155427926685247020589, −5.68504934958089764262672199406, −4.32649131554087306140132486420, −3.20259660323436268854874498903, −2.39889917273376120677588104461, 0.69293433755956247229683301550, 1.90284611735174647136568173103, 3.90922605867524494959903509466, 4.95074603585306449643708346454, 6.27782440392599773076516055581, 7.54223973649710689887818939125, 7.86378415301078788630660370981, 8.759328025962668525246036663052, 9.373825422312231493155042799556, 10.77580621046470573427793974231

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