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Properties

Label 2-462-33.32-c1-0-23
Degree $2$
Conductor $462$
Sign $0.0628 + 0.998i$
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 + (0.806 − 1.53i)3-s + 4-s − 3.86i·5-s + (0.806 − 1.53i)6-s + i·7-s + 8-s + (−1.69 − 2.47i)9-s − 3.86i·10-s + (1.72 + 2.83i)11-s + (0.806 − 1.53i)12-s + 5.06i·13-s + i·14-s + (−5.91 − 3.11i)15-s + 16-s − 2.69·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.465 − 0.884i)3-s + 0.5·4-s − 1.72i·5-s + (0.329 − 0.625i)6-s + 0.377i·7-s + 0.353·8-s + (−0.566 − 0.824i)9-s − 1.22i·10-s + (0.520 + 0.853i)11-s + (0.232 − 0.442i)12-s + 1.40i·13-s + 0.267i·14-s + (−1.52 − 0.804i)15-s + 0.250·16-s − 0.654·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0628 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0628 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.0628 + 0.998i$
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: \((2,\ 462,\ (\ :1/2),\ 0.0628 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76226 - 1.65469i\)
\(L(\frac12)\) \(\approx\) \(1.76226 - 1.65469i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.806 + 1.53i)T \)
7 \( 1 - iT \)
11 \( 1 + (-1.72 - 2.83i)T \)
good5 \( 1 + 3.86iT - 5T^{2} \)
13 \( 1 - 5.06iT - 13T^{2} \)
17 \( 1 + 2.69T + 17T^{2} \)
19 \( 1 + 7.70iT - 19T^{2} \)
23 \( 1 - 3.49iT - 23T^{2} \)
29 \( 1 + 0.831T + 29T^{2} \)
31 \( 1 - 9.19T + 31T^{2} \)
37 \( 1 - 3.04T + 37T^{2} \)
41 \( 1 - 5.28T + 41T^{2} \)
43 \( 1 - 12.5iT - 43T^{2} \)
47 \( 1 + 4.96iT - 47T^{2} \)
53 \( 1 + 0.602iT - 53T^{2} \)
59 \( 1 - 0.161iT - 59T^{2} \)
61 \( 1 - 4.61iT - 61T^{2} \)
67 \( 1 + 4.22T + 67T^{2} \)
71 \( 1 + 12.0iT - 71T^{2} \)
73 \( 1 - 6.15iT - 73T^{2} \)
79 \( 1 + 2.49iT - 79T^{2} \)
83 \( 1 + 3.53T + 83T^{2} \)
89 \( 1 - 7.35iT - 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−11.53343901305461024228028030763, −9.384319767163508523324868570348, −9.152966284601683222922038643516, −8.120345376851862349249872694445, −7.01759041183413623144468235809, −6.20231632368952709976425158188, −4.85115369214287744664939177753, −4.25784199598278947502469639771, −2.44063331595725408448409683589, −1.32497047885403929937003078545, 2.58265109647525417213385090668, 3.37127757811482765899624383938, 4.14136486985138713197764983901, 5.68997245516043420386560331853, 6.39632914011894283358107661020, 7.60576386370401981913442333406, 8.388060082421366984520979683166, 9.914474042938967417255855588966, 10.52599783543627329668446062113, 10.97089807178870869254585563083

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