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Properties

Label 2-462-7.4-c1-0-5
Degree $2$
Conductor $462$
Sign $0.0633 - 0.997i$
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  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (1.32 + 2.29i)5-s + 0.999·6-s + (1.32 + 2.29i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.32 + 2.29i)10-s + (−0.5 + 0.866i)11-s + (0.499 + 0.866i)12-s − 4·13-s + (−1.32 + 2.29i)14-s + 2.64·15-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.591 + 1.02i)5-s + 0.408·6-s + (0.499 + 0.866i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.418 + 0.724i)10-s + (−0.150 + 0.261i)11-s + (0.144 + 0.249i)12-s − 1.10·13-s + (−0.353 + 0.612i)14-s + 0.683·15-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.0633 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41385 + 1.32697i\)
\(L(\frac12)\) \(\approx\) \(1.41385 + 1.32697i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.32 - 2.29i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-1.32 - 2.29i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.64 - 4.58i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.32 + 2.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.64 - 8.04i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + 1.29T + 43T^{2} \)
47 \( 1 + (5.96 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.29 + 12.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.96 - 3.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.79 + 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + (-2.35 + 4.07i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.67 + 4.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.58T + 83T^{2} \)
89 \( 1 + (6.64 + 11.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.58T + 97T^{2} \)
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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

−11.56411505624393456927418841752, −10.14642876048558189980020572901, −9.513568098854201167122386152759, −8.230966938238508449746171483664, −7.55903097165299047509326303899, −6.60301160656950150316639174178, −5.78614763359669439575931096608, −4.77118016634481295555863848796, −3.05726337605455684536883130590, −2.19737685816169299575441265046, 1.15323893719276258549884263292, 2.65749543052844800529698546142, 4.10347119185962035858682496663, 4.88220533494781144924774000312, 5.66738469279456934503339424824, 7.27520420822457948050111576567, 8.316110735535002883244964476541, 9.335051793121086217075351010748, 9.897493302347838386153429968078, 10.81098707389742647753365663352

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