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Properties

Label 2-462-77.62-c1-0-8
Degree $2$
Conductor $462$
Sign $0.995 + 0.0941i$
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  + (0.587 − 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (1.08 + 1.49i)5-s + (0.809 − 0.587i)6-s + (−2.23 + 1.41i)7-s + (−0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + 1.85·10-s + (3.19 + 0.905i)11-s − i·12-s + (3.89 + 2.83i)13-s + (−0.165 + 2.64i)14-s + (0.572 + 1.76i)15-s + (−0.809 + 0.587i)16-s + (4.59 − 3.34i)17-s + ⋯
L(s)  = 1  + (0.415 − 0.572i)2-s + (0.549 + 0.178i)3-s + (−0.154 − 0.475i)4-s + (0.486 + 0.669i)5-s + (0.330 − 0.239i)6-s + (−0.844 + 0.536i)7-s + (−0.336 − 0.109i)8-s + (0.269 + 0.195i)9-s + 0.585·10-s + (0.962 + 0.272i)11-s − 0.288i·12-s + (1.08 + 0.784i)13-s + (−0.0441 + 0.705i)14-s + (0.147 + 0.454i)15-s + (−0.202 + 0.146i)16-s + (1.11 − 0.810i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.995 + 0.0941i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.995 + 0.0941i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16364 - 0.102101i\)
\(L(\frac12)\) \(\approx\) \(2.16364 - 0.102101i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 + 0.809i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (2.23 - 1.41i)T \)
11 \( 1 + (-3.19 - 0.905i)T \)
good5 \( 1 + (-1.08 - 1.49i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-3.89 - 2.83i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.59 + 3.34i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.53 + 4.73i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.28T + 23T^{2} \)
29 \( 1 + (7.07 - 2.29i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.47 - 7.54i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.14 - 3.53i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.770 - 2.37i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 11.6iT - 43T^{2} \)
47 \( 1 + (1.28 + 0.416i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.64 + 6.28i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (6.16 - 2.00i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-9.01 + 6.54i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 0.825T + 67T^{2} \)
71 \( 1 + (4.25 - 3.09i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.97 - 6.08i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.82 + 10.7i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.22 - 5.97i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 9.76iT - 89T^{2} \)
97 \( 1 + (10.0 - 13.8i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.09502957578820082724048078028, −10.02529594485075747333152169929, −9.401869242933821646900777193684, −8.743930529307066966823818001254, −7.08871872193864729198545824886, −6.38529777782388342700782342112, −5.27871231917227946792767700943, −3.79673150232722192722819093323, −3.08029019071862823721534758387, −1.80428799403313862367294788618, 1.39902853526712391568507184252, 3.46224824882426266957855703686, 3.96394555172340599391130711547, 5.84907694923246091075478256605, 6.02825862246139906753346555644, 7.52669508096469146711395442989, 8.167160423640705563085501034217, 9.282059015789026453946077154213, 9.823824800281884203803878062903, 11.08182449326068364076097526590

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