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Properties

Label 2-462-33.29-c1-0-11
Degree $2$
Conductor $462$
Sign $0.543 + 0.839i$
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  + (−0.809 − 0.587i)2-s + (0.261 − 1.71i)3-s + (0.309 + 0.951i)4-s + (1.43 + 1.97i)5-s + (−1.21 + 1.23i)6-s + (−0.951 + 0.309i)7-s + (0.309 − 0.951i)8-s + (−2.86 − 0.896i)9-s − 2.44i·10-s + (1.53 + 2.93i)11-s + (1.70 − 0.280i)12-s + (4.02 − 5.54i)13-s + (0.951 + 0.309i)14-s + (3.76 − 1.94i)15-s + (−0.809 + 0.587i)16-s + (1.36 − 0.993i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.151 − 0.988i)3-s + (0.154 + 0.475i)4-s + (0.642 + 0.884i)5-s + (−0.497 + 0.502i)6-s + (−0.359 + 0.116i)7-s + (0.109 − 0.336i)8-s + (−0.954 − 0.298i)9-s − 0.773i·10-s + (0.462 + 0.886i)11-s + (0.493 − 0.0808i)12-s + (1.11 − 1.53i)13-s + (0.254 + 0.0825i)14-s + (0.971 − 0.501i)15-s + (−0.202 + 0.146i)16-s + (0.331 − 0.240i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.543 + 0.839i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11354 - 0.605915i\)
\(L(\frac12)\) \(\approx\) \(1.11354 - 0.605915i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-0.261 + 1.71i)T \)
7 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (-1.53 - 2.93i)T \)
good5 \( 1 + (-1.43 - 1.97i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-4.02 + 5.54i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.36 + 0.993i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-6.15 - 1.99i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.69iT - 23T^{2} \)
29 \( 1 + (-1.73 - 5.32i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.638 + 0.463i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.12 - 6.52i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.79 + 8.59i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.16iT - 43T^{2} \)
47 \( 1 + (4.81 + 1.56i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.765 + 1.05i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (13.0 - 4.22i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.87 - 8.09i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 1.03T + 67T^{2} \)
71 \( 1 + (4.46 + 6.14i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.108 + 0.0352i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.14 - 8.45i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.09 - 1.52i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 11.7iT - 89T^{2} \)
97 \( 1 + (-0.828 - 0.601i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.70208168790179544130350369633, −10.16284058707528691468350459088, −9.128504424520678331969738735289, −8.167882755734978108172940306940, −7.25188382407525498723881041766, −6.48427928259214982009116482493, −5.54883356577079722538012207003, −3.40121043828378558491590102490, −2.60801791301847592630066776961, −1.19368127129949435594699163817, 1.35678173484312992917565422597, 3.33933230544263257577721792191, 4.51808989333239528746776931986, 5.67016958092077473762773306060, 6.30282947441184327135991071851, 7.80502767418817899113032555562, 8.874061459778182062834185052484, 9.318360789217588326070431808959, 9.846453405875471978546139941632, 11.24836294100992965011755190727

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