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Properties

Label 2-462-21.5-c1-0-23
Degree $2$
Conductor $462$
Sign $0.832 - 0.553i$
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 + (2.59 − 0.499i)14-s + 6·15-s + (−0.5 + 0.866i)16-s + (2.59 + 4.5i)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 + (0.694 − 0.133i)14-s + 1.54·15-s + (−0.125 + 0.216i)16-s + (0.630 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.832 - 0.553i$
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: \((2,\ 462,\ (\ :1/2),\ 0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50476 + 0.756976i\)
\(L(\frac12)\) \(\approx\) \(2.50476 + 0.756976i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 4.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.5 + 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 9iT - 29T^{2} \)
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 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (6.06 - 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9iT - 71T^{2} \)
73 \( 1 + (-12 + 6.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 + (-1.73 + 3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.73iT - 97T^{2} \)
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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

−10.82617197121188793976057077221, −10.37600182271168175810749611753, −9.194768342564215755557158794624, −8.376028884334979737863144686848, −7.79104703082030331581357795522, −6.13036546378618406911746083345, −5.04579616750477943199002335680, −4.68523383637415143507961795950, −3.42910559444534169949543933545, −1.76415532149147300528373241422, 1.99848658964301195000286311142, 2.48779564962119649718578641349, 3.81991190187978153851156753022, 5.50084225807538875901021501297, 6.26006777481575583679881815968, 7.12306365377548817939191212871, 8.093091745409264097474413296750, 9.301309176794496312837592430595, 10.13374502267682623562165740709, 11.30702604956726645852248933462

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