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Properties

Label 2-462-11.4-c1-0-8
Degree $2$
Conductor $462$
Sign $0.167 + 0.985i$
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  + (−0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (2.52 − 1.83i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 3.11·10-s + (2.97 − 1.46i)11-s + 12-s + (1.74 + 1.27i)13-s + (0.309 − 0.951i)14-s + (−0.962 − 2.96i)15-s + (−0.809 + 0.587i)16-s + (−0.574 + 0.417i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 + 0.475i)4-s + (1.12 − 0.818i)5-s + (−0.330 + 0.239i)6-s + (0.116 + 0.359i)7-s + (0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s − 0.985·10-s + (0.896 − 0.442i)11-s + 0.288·12-s + (0.484 + 0.352i)13-s + (0.0825 − 0.254i)14-s + (−0.248 − 0.764i)15-s + (−0.202 + 0.146i)16-s + (−0.139 + 0.101i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.167 + 0.985i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.167 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08776 - 0.918743i\)
\(L(\frac12)\) \(\approx\) \(1.08776 - 0.918743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-2.97 + 1.46i)T \)
good5 \( 1 + (-2.52 + 1.83i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.74 - 1.27i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.574 - 0.417i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.0949 + 0.292i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.879T + 23T^{2} \)
29 \( 1 + (-2.03 - 6.25i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.21 + 4.51i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.19 + 3.68i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.32 + 4.07i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.24T + 43T^{2} \)
47 \( 1 + (-3.10 + 9.54i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.32 - 3.86i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.77 + 5.46i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.52 - 1.83i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.67T + 67T^{2} \)
71 \( 1 + (8.77 - 6.37i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.32 - 10.2i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.86 - 5.71i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.7 - 9.27i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 + (-0.211 - 0.153i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.85733420422956263572471117998, −9.738804302780582481329565531848, −8.843607512298179741159585147098, −8.689397773173797152975724717627, −7.21302223580013869614563540038, −6.22079330406422725412275825665, −5.29145950123176873204807834418, −3.72547445798622562159639928213, −2.17005632878693401545062499304, −1.23227677090498508388998701656, 1.71554922661605547441226596391, 3.15856692873487934381894599146, 4.59404907704110533908284828906, 5.88441712026310719667729116578, 6.57392875380844617706450666535, 7.56001447308581008607737025768, 8.752584328040672054973622361798, 9.530670846476527063285798392865, 10.22694381391199343572168085153, 10.83549404478609243759259467951

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