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Properties

Label 2-462-77.13-c1-0-10
Degree $2$
Conductor $462$
Sign $-0.344 + 0.938i$
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  + (−0.951 − 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (0.0789 − 0.0256i)5-s + (0.309 + 0.951i)6-s + (−1.88 + 1.86i)7-s + (−0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s − 0.0830·10-s + (2.36 − 2.32i)11-s − 0.999i·12-s + (0.764 − 2.35i)13-s + (2.36 − 1.18i)14-s + (−0.0672 − 0.0488i)15-s + (0.309 + 0.951i)16-s + (−1.92 − 5.92i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (−0.339 − 0.467i)3-s + (0.404 + 0.293i)4-s + (0.0353 − 0.0114i)5-s + (0.126 + 0.388i)6-s + (−0.710 + 0.703i)7-s + (−0.207 − 0.286i)8-s + (−0.103 + 0.317i)9-s − 0.0262·10-s + (0.714 − 0.699i)11-s − 0.288i·12-s + (0.211 − 0.652i)13-s + (0.631 − 0.317i)14-s + (−0.0173 − 0.0126i)15-s + (0.0772 + 0.237i)16-s + (−0.466 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.344 + 0.938i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.344 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396712 - 0.568316i\)
\(L(\frac12)\) \(\approx\) \(0.396712 - 0.568316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (1.88 - 1.86i)T \)
11 \( 1 + (-2.36 + 2.32i)T \)
good5 \( 1 + (-0.0789 + 0.0256i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.764 + 2.35i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.92 + 5.92i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.47 + 2.52i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.15T + 23T^{2} \)
29 \( 1 + (-4.17 + 5.74i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (5.72 + 1.86i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.56 + 2.59i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.41 + 6.84i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.35iT - 43T^{2} \)
47 \( 1 + (7.52 + 10.3i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.26 + 3.88i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.91 - 8.13i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.85 + 5.71i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + (1.82 + 5.63i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.23 + 2.34i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-10.7 - 3.48i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.02 - 12.3i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 11.5iT - 89T^{2} \)
97 \( 1 + (-16.1 - 5.23i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.93767913232074034568954089109, −9.614643411136406738549244786615, −9.195415555473310895937110210710, −8.098692459507795353510738081282, −7.11989195195208615838216842707, −6.21923095236806725531953630522, −5.32248187358376716845345780366, −3.50994021827810197143806297434, −2.37218059078467799001341602447, −0.57291832674822484908381840094, 1.56439479097216950682429052847, 3.53852593448319213772764392362, 4.49000238856385903042788350780, 6.02010126028821147505868737746, 6.65425373572703087068277895974, 7.64258942302026379842683822573, 8.825497686330183053450525089981, 9.617555675202156005740483277536, 10.28737182888568906688208974830, 11.04334738999520454296209525942

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