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Properties

Label 2-462-33.32-c1-0-17
Degree $2$
Conductor $462$
Sign $0.453 + 0.891i$
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  − 2-s + (1.37 + 1.05i)3-s + 4-s − 3.92i·5-s + (−1.37 − 1.05i)6-s − i·7-s − 8-s + (0.775 + 2.89i)9-s + 3.92i·10-s + (1.42 − 2.99i)11-s + (1.37 + 1.05i)12-s + 0.109i·13-s + i·14-s + (4.14 − 5.39i)15-s + 16-s − 6.97·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.793 + 0.608i)3-s + 0.5·4-s − 1.75i·5-s + (−0.560 − 0.430i)6-s − 0.377i·7-s − 0.353·8-s + (0.258 + 0.965i)9-s + 1.24i·10-s + (0.430 − 0.902i)11-s + (0.396 + 0.304i)12-s + 0.0302i·13-s + 0.267i·14-s + (1.06 − 1.39i)15-s + 0.250·16-s − 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.453 + 0.891i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.453 + 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06589 - 0.653443i\)
\(L(\frac12)\) \(\approx\) \(1.06589 - 0.653443i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.37 - 1.05i)T \)
7 \( 1 + iT \)
11 \( 1 + (-1.42 + 2.99i)T \)
good5 \( 1 + 3.92iT - 5T^{2} \)
13 \( 1 - 0.109iT - 13T^{2} \)
17 \( 1 + 6.97T + 17T^{2} \)
19 \( 1 + 5.78iT - 19T^{2} \)
23 \( 1 + 0.938iT - 23T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 + 2.61T + 31T^{2} \)
37 \( 1 - 7.21T + 37T^{2} \)
41 \( 1 - 8.95T + 41T^{2} \)
43 \( 1 + 1.44iT - 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 + 5.55iT - 53T^{2} \)
59 \( 1 - 7.60iT - 59T^{2} \)
61 \( 1 - 6.35iT - 61T^{2} \)
67 \( 1 - 5.60T + 67T^{2} \)
71 \( 1 + 4.39iT - 71T^{2} \)
73 \( 1 + 3.31iT - 73T^{2} \)
79 \( 1 - 0.361iT - 79T^{2} \)
83 \( 1 + 1.48T + 83T^{2} \)
89 \( 1 + 0.722iT - 89T^{2} \)
97 \( 1 + 3.48T + 97T^{2} \)
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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

−10.80699941982024120213924739340, −9.626683057499041793182432623913, −8.861675733865629220617833502167, −8.679458995771583348532516721360, −7.64182644704391812236607161865, −6.29232061044267350968181987785, −4.83222932299571880400342738382, −4.19873033015271524613059330128, −2.55686605449881183096465074991, −0.914638940317646164890418469371, 1.99381967249968696485408618033, 2.75845745771989833176314290614, 4.00666810879765836097574170679, 6.23480948544139299712230308799, 6.74073919816005306847516763988, 7.53397322521884640581368498145, 8.410999689981672027953446234417, 9.470030156671747162927391147157, 10.15351042438030445906529347604, 11.10100062465345080830420056510

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