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Properties

Label 2-462-77.62-c1-0-2
Degree $2$
Conductor $462$
Sign $0.708 - 0.705i$
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 + (2.18 + 3.01i)5-s + (−0.809 + 0.587i)6-s + (−2.24 + 1.39i)7-s + (−0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + 3.72·10-s + (−2.71 + 1.90i)11-s + i·12-s + (4.10 + 2.98i)13-s + (−0.190 + 2.63i)14-s + (−1.14 − 3.53i)15-s + (−0.809 + 0.587i)16-s + (−4.90 + 3.56i)17-s + ⋯
L(s)  = 1  + (0.415 − 0.572i)2-s + (−0.549 − 0.178i)3-s + (−0.154 − 0.475i)4-s + (0.978 + 1.34i)5-s + (−0.330 + 0.239i)6-s + (−0.849 + 0.528i)7-s + (−0.336 − 0.109i)8-s + (0.269 + 0.195i)9-s + 1.17·10-s + (−0.819 + 0.573i)11-s + 0.288i·12-s + (1.13 + 0.828i)13-s + (−0.0508 + 0.705i)14-s + (−0.296 − 0.913i)15-s + (−0.202 + 0.146i)16-s + (−1.18 + 0.864i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.708 - 0.705i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.708 - 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26720 + 0.523594i\)
\(L(\frac12)\) \(\approx\) \(1.26720 + 0.523594i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.24 - 1.39i)T \)
11 \( 1 + (2.71 - 1.90i)T \)
good5 \( 1 + (-2.18 - 3.01i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-4.10 - 2.98i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.90 - 3.56i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.0255 + 0.0785i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.24T + 23T^{2} \)
29 \( 1 + (-9.40 + 3.05i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.60 + 2.20i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.49 + 7.68i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.40 - 7.41i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.56iT - 43T^{2} \)
47 \( 1 + (-9.34 - 3.03i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.23 - 0.894i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.05 + 0.993i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.40 - 2.47i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.56T + 67T^{2} \)
71 \( 1 + (-5.72 + 4.16i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.831 + 2.55i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.41 - 8.82i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.69 - 7.04i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 + (-4.43 + 6.09i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.97991977711185994049481220750, −10.50247175943043047900674681531, −9.735772767152685273959781666998, −8.717376364808819097031749016925, −6.99174730059531160584612710511, −6.33420641835241040554801348406, −5.76008220081038066007048669167, −4.28739732138796164574361501899, −2.88120791516342532163556141880, −2.01677940332408079954031497367, 0.801135801235629177758081535201, 3.02601219853583844067805140388, 4.50594229733742533458600739140, 5.30363844685602199343015880116, 6.07216357550333806313779643775, 6.93023725037739569002800294629, 8.421450543356873716272207716043, 8.950767461200545605050204079530, 10.09093941941518868943796869498, 10.78081068210551076812870806003

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