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Properties

Label 2-462-77.76-c1-0-14
Degree $2$
Conductor $462$
Sign $-0.975 + 0.221i$
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  − i·2-s + i·3-s − 4-s − 4.14i·5-s + 6-s + (−0.717 − 2.54i)7-s + i·8-s − 9-s − 4.14·10-s + (−0.170 + 3.31i)11-s − i·12-s − 3.09·13-s + (−2.54 + 0.717i)14-s + 4.14·15-s + 16-s − 3.57·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 1.85i·5-s + 0.408·6-s + (−0.271 − 0.962i)7-s + 0.353i·8-s − 0.333·9-s − 1.30·10-s + (−0.0514 + 0.998i)11-s − 0.288i·12-s − 0.857·13-s + (−0.680 + 0.191i)14-s + 1.06·15-s + 0.250·16-s − 0.867·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.975 + 0.221i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.975 + 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0947383 - 0.845849i\)
\(L(\frac12)\) \(\approx\) \(0.0947383 - 0.845849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - iT \)
7 \( 1 + (0.717 + 2.54i)T \)
11 \( 1 + (0.170 - 3.31i)T \)
good5 \( 1 + 4.14iT - 5T^{2} \)
13 \( 1 + 3.09T + 13T^{2} \)
17 \( 1 + 3.57T + 17T^{2} \)
19 \( 1 + 3.91T + 19T^{2} \)
23 \( 1 - 7.71T + 23T^{2} \)
29 \( 1 + 1.65iT - 29T^{2} \)
31 \( 1 + 9.23iT - 31T^{2} \)
37 \( 1 + 0.869T + 37T^{2} \)
41 \( 1 - 5.91T + 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 + 6.76iT - 47T^{2} \)
53 \( 1 + 1.88T + 53T^{2} \)
59 \( 1 - 10.1iT - 59T^{2} \)
61 \( 1 - 3.37T + 61T^{2} \)
67 \( 1 - 9.84T + 67T^{2} \)
71 \( 1 - 1.33T + 71T^{2} \)
73 \( 1 - 5.57T + 73T^{2} \)
79 \( 1 - 10.8iT - 79T^{2} \)
83 \( 1 - 5.67T + 83T^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 + 4.52iT - 97T^{2} \)
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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

−10.54257833712097059037970890965, −9.647334171712923132243654902461, −9.167260117057530386946206006461, −8.216737142181281141903231035263, −7.06686082995757851696837727550, −5.39376829873689902852770346964, −4.51005587222387571875609977946, −4.09077513767517243708847802896, −2.16469668894171566408164273937, −0.50747095365567883428861397690, 2.49042581092068908437603859669, 3.27301771858111894207809962930, 5.10297093563442731147170217499, 6.38598412729571679481471606742, 6.58727534657912631257134661810, 7.60958595791275466490976105962, 8.615884373899512679562368356939, 9.515407501647573753103777198596, 10.79926990426950412519783555270, 11.22590733238202886672097479467

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