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Properties

Label 2-462-33.17-c1-0-19
Degree $2$
Conductor $462$
Sign $0.719 - 0.694i$
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  + (0.309 + 0.951i)2-s + (1.65 − 0.520i)3-s + (−0.809 + 0.587i)4-s + (2.53 + 0.823i)5-s + (1.00 + 1.41i)6-s + (−0.587 − 0.809i)7-s + (−0.809 − 0.587i)8-s + (2.45 − 1.71i)9-s + 2.66i·10-s + (−2.99 + 1.41i)11-s + (−1.03 + 1.39i)12-s + (4.96 − 1.61i)13-s + (0.587 − 0.809i)14-s + (4.61 + 0.0410i)15-s + (0.309 − 0.951i)16-s + (−2.02 + 6.22i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.953 − 0.300i)3-s + (−0.404 + 0.293i)4-s + (1.13 + 0.368i)5-s + (0.410 + 0.575i)6-s + (−0.222 − 0.305i)7-s + (−0.286 − 0.207i)8-s + (0.819 − 0.573i)9-s + 0.842i·10-s + (−0.904 + 0.426i)11-s + (−0.297 + 0.401i)12-s + (1.37 − 0.447i)13-s + (0.157 − 0.216i)14-s + (1.19 + 0.0106i)15-s + (0.0772 − 0.237i)16-s + (−0.490 + 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.719 - 0.694i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.719 - 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17643 + 0.879492i\)
\(L(\frac12)\) \(\approx\) \(2.17643 + 0.879492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (-1.65 + 0.520i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (2.99 - 1.41i)T \)
good5 \( 1 + (-2.53 - 0.823i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (-4.96 + 1.61i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.02 - 6.22i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.228 + 0.314i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 7.82iT - 23T^{2} \)
29 \( 1 + (8.16 - 5.93i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.410 + 1.26i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.989 + 0.718i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.86 - 1.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.75iT - 43T^{2} \)
47 \( 1 + (4.74 - 6.53i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-6.70 + 2.17i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.34 - 3.22i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (9.02 + 2.93i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 7.83T + 67T^{2} \)
71 \( 1 + (8.10 + 2.63i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.44 + 4.74i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.34 + 0.435i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.943 + 2.90i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6.89iT - 89T^{2} \)
97 \( 1 + (1.15 + 3.54i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.74439720208423275696991187089, −10.25294926808149110016308582662, −9.123972550532472862922386838109, −8.418749237021274974157285718052, −7.49312764416476373759326302086, −6.45159436806553499000751080372, −5.84204059953602181573322738636, −4.31028481794400934255056137120, −3.14295764235949942334751028673, −1.85491062596962511010939404562, 1.67828702118152738571495161532, 2.73197029834053122280560124428, 3.81632647560661204870383770151, 5.15880438683851451190903598204, 5.92328755486614914606087544909, 7.44239501127655030953743360949, 8.634923375014153750635142471806, 9.336834472598083707349713941671, 9.803358200972792367181850484687, 10.89468838308368929124888624139

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