Properties

Label 2-462-7.4-c1-0-7
Degree $2$
Conductor $462$
Sign $-0.480 + 0.877i$
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 + (−0.227 − 0.393i)5-s + 0.999·6-s + (−0.227 − 2.63i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.227 + 0.393i)10-s + (−0.5 + 0.866i)11-s + (−0.499 − 0.866i)12-s + 0.909·13-s + (−2.16 + 1.51i)14-s + 0.454·15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.101 − 0.176i)5-s + 0.408·6-s + (−0.0859 − 0.996i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.0719 + 0.124i)10-s + (−0.150 + 0.261i)11-s + (−0.144 − 0.249i)12-s + 0.252·13-s + (−0.579 + 0.404i)14-s + 0.117·15-s + (−0.125 − 0.216i)16-s + (−0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.480 + 0.877i$
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: \((2,\ 462,\ (\ :1/2),\ -0.480 + 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.376584 - 0.635496i\)
\(L(\frac12)\) \(\approx\) \(0.376584 - 0.635496i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.227 + 2.63i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.227 + 0.393i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 0.909T + 13T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.94 + 3.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.16 + 7.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.79T + 29T^{2} \)
31 \( 1 + (-4.79 + 8.30i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.94 + 6.82i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.79T + 41T^{2} \)
43 \( 1 - 7.88T + 43T^{2} \)
47 \( 1 + (0.169 + 0.292i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.454 - 0.787i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.48 - 6.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.02 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.89 + 3.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.79T + 71T^{2} \)
73 \( 1 + (5.33 - 9.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.68 + 4.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.97T + 83T^{2} \)
89 \( 1 + (-5.54 - 9.60i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−10.58371292130581365900565632347, −10.16770966135559362284045564044, −9.093834079837554598264789420524, −8.231759369692251454082449355827, −7.16374600856770660544118281105, −6.06595842570451276506128523729, −4.52377027537023405314432433465, −4.02489999895463392542565645755, −2.46471176577586186006788628156, −0.53719408515018209833302588350, 1.70601479371730345511682395519, 3.33158883486157952311631093762, 5.05263811720458959983837046862, 5.87088351832905122411404056090, 6.67713450092300723062095075313, 7.73753851721326279164878502037, 8.498423017258260570315522193678, 9.379926216460516368183928720843, 10.39702717104053530987053348281, 11.39906557944883940293475129283

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