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Properties

Label 2-462-77.41-c1-0-7
Degree $2$
Conductor $462$
Sign $0.879 + 0.475i$
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  + (−0.587 − 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−1.12 + 1.54i)5-s + (−0.809 − 0.587i)6-s + (1.92 − 1.81i)7-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + 1.90·10-s + (−0.215 + 3.30i)11-s + i·12-s + (2.19 − 1.59i)13-s + (−2.60 − 0.488i)14-s + (−0.589 + 1.81i)15-s + (−0.809 − 0.587i)16-s + (3.34 + 2.42i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (0.549 − 0.178i)3-s + (−0.154 + 0.475i)4-s + (−0.501 + 0.690i)5-s + (−0.330 − 0.239i)6-s + (0.726 − 0.686i)7-s + (0.336 − 0.109i)8-s + (0.269 − 0.195i)9-s + 0.603·10-s + (−0.0648 + 0.997i)11-s + 0.288i·12-s + (0.610 − 0.443i)13-s + (−0.694 − 0.130i)14-s + (−0.152 + 0.468i)15-s + (−0.202 − 0.146i)16-s + (0.810 + 0.588i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35778 - 0.343144i\)
\(L(\frac12)\) \(\approx\) \(1.35778 - 0.343144i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 + (-1.92 + 1.81i)T \)
11 \( 1 + (0.215 - 3.30i)T \)
good5 \( 1 + (1.12 - 1.54i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-2.19 + 1.59i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.34 - 2.42i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.23 + 3.79i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.88T + 23T^{2} \)
29 \( 1 + (-6.78 - 2.20i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.61 + 2.22i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.321 + 0.988i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.51 + 7.73i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.21iT - 43T^{2} \)
47 \( 1 + (7.21 - 2.34i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.96 - 2.88i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.13 + 1.01i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.21 - 4.51i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 1.75T + 67T^{2} \)
71 \( 1 + (10.1 + 7.36i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.13 - 6.57i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.00 + 1.38i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.15 + 0.839i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.07iT - 89T^{2} \)
97 \( 1 + (3.78 + 5.21i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.80654240227966080394577168286, −10.33874991646253331714318442948, −9.173763967046815879736579993854, −8.235068939873769122954347643799, −7.46363427943305493978473899045, −6.79917853165309441656748230574, −4.94108431000111941319263221804, −3.82777606555839371444040135065, −2.83156315185505055982443473923, −1.33678529913375776636177433281, 1.27650590623926293310505425997, 3.11083807805815076140356646307, 4.52879599946273282939977306536, 5.39359346556520207976736674592, 6.53474135406988440121536627254, 7.88709997895269746907489695077, 8.427304884177774564045475000136, 8.930578710452829632994388272615, 10.00626627078353967862700531765, 11.13067983480748579438245787405

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