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Properties

Label 2-462-21.5-c1-0-15
Degree $2$
Conductor $462$
Sign $0.0285 - 0.999i$
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  + (0.866 + 0.5i)2-s + (1.62 + 0.599i)3-s + (0.499 + 0.866i)4-s + (−0.758 + 1.31i)5-s + (1.10 + 1.33i)6-s + (−1.48 + 2.19i)7-s + 0.999i·8-s + (2.28 + 1.94i)9-s + (−1.31 + 0.758i)10-s + (0.866 − 0.5i)11-s + (0.292 + 1.70i)12-s − 5.44i·13-s + (−2.38 + 1.15i)14-s + (−2.02 + 1.68i)15-s + (−0.5 + 0.866i)16-s + (2.34 + 4.05i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.938 + 0.346i)3-s + (0.249 + 0.433i)4-s + (−0.339 + 0.587i)5-s + (0.452 + 0.543i)6-s + (−0.560 + 0.827i)7-s + 0.353i·8-s + (0.760 + 0.649i)9-s + (−0.415 + 0.239i)10-s + (0.261 − 0.150i)11-s + (0.0845 + 0.492i)12-s − 1.51i·13-s + (−0.636 + 0.308i)14-s + (−0.521 + 0.433i)15-s + (−0.125 + 0.216i)16-s + (0.567 + 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0285 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0285 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.0285 - 0.999i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.0285 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72589 + 1.67738i\)
\(L(\frac12)\) \(\approx\) \(1.72589 + 1.67738i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-1.62 - 0.599i)T \)
7 \( 1 + (1.48 - 2.19i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (0.758 - 1.31i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 5.44iT - 13T^{2} \)
17 \( 1 + (-2.34 - 4.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.12 + 3.53i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.680 - 0.392i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.48iT - 29T^{2} \)
31 \( 1 + (-7.11 + 4.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.996 + 1.72i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.74T + 41T^{2} \)
43 \( 1 - 2.14T + 43T^{2} \)
47 \( 1 + (-1.58 + 2.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.99 - 2.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.55 + 4.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.29 + 4.78i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.83 - 6.64i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.76iT - 71T^{2} \)
73 \( 1 + (-8.74 + 5.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.25 + 2.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 + (4.91 - 8.50i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.37iT - 97T^{2} \)
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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

−11.16981713955604392460135776793, −10.41634334211789237736767345428, −9.373471543751228920826860629442, −8.373072090456461082974609161206, −7.75747808390312482805354463737, −6.55797680217682213019847661575, −5.66292029034721628720878513452, −4.31488317163077542132612395443, −3.26813151777083479555358025680, −2.53039338069320344196744576661, 1.27805452504667719218368308127, 2.75174334809790586230702843021, 4.03544662015171702535915787234, 4.52316448818558919118572317589, 6.36989861640064587955760705677, 7.02806830146687385258685295320, 8.120531267195359563928050879397, 9.129200901857367456445049182827, 9.835721251111839532247369404124, 10.86994947366266127706591426857

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