Properties

Label 2-462-77.16-c1-0-1
Degree $2$
Conductor $462$
Sign $0.980 - 0.198i$
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  + (−0.978 − 0.207i)2-s + (−0.104 + 0.994i)3-s + (0.913 + 0.406i)4-s + (−2.71 − 3.02i)5-s + (0.309 − 0.951i)6-s + (−1.98 + 1.75i)7-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (2.03 + 3.51i)10-s + (2.84 + 1.70i)11-s + (−0.5 + 0.866i)12-s + (0.914 + 2.81i)13-s + (2.30 − 1.30i)14-s + (3.28 − 2.38i)15-s + (0.669 + 0.743i)16-s + (4.37 − 0.929i)17-s + ⋯
L(s)  = 1  + (−0.691 − 0.147i)2-s + (−0.0603 + 0.574i)3-s + (0.456 + 0.203i)4-s + (−1.21 − 1.35i)5-s + (0.126 − 0.388i)6-s + (−0.749 + 0.661i)7-s + (−0.286 − 0.207i)8-s + (−0.326 − 0.0693i)9-s + (0.642 + 1.11i)10-s + (0.857 + 0.514i)11-s + (−0.144 + 0.250i)12-s + (0.253 + 0.780i)13-s + (0.615 − 0.347i)14-s + (0.848 − 0.616i)15-s + (0.167 + 0.185i)16-s + (1.06 − 0.225i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.980 - 0.198i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.980 - 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.770122 + 0.0773422i\)
\(L(\frac12)\) \(\approx\) \(0.770122 + 0.0773422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.978 + 0.207i)T \)
3 \( 1 + (0.104 - 0.994i)T \)
7 \( 1 + (1.98 - 1.75i)T \)
11 \( 1 + (-2.84 - 1.70i)T \)
good5 \( 1 + (2.71 + 3.02i)T + (-0.522 + 4.97i)T^{2} \)
13 \( 1 + (-0.914 - 2.81i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.37 + 0.929i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (-6.68 + 2.97i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (-1.81 + 3.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.324 - 0.235i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.09 + 4.54i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-0.904 - 8.60i)T + (-36.1 + 7.69i)T^{2} \)
41 \( 1 + (-5.50 - 3.99i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.88T + 43T^{2} \)
47 \( 1 + (0.119 - 0.0530i)T + (31.4 - 34.9i)T^{2} \)
53 \( 1 + (0.0143 - 0.0159i)T + (-5.54 - 52.7i)T^{2} \)
59 \( 1 + (2.86 + 1.27i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (5.14 + 5.71i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (-2.73 - 4.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.19 - 12.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.02 - 1.34i)T + (48.8 + 54.2i)T^{2} \)
79 \( 1 + (-16.3 - 3.46i)T + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (-2.23 + 6.88i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (0.223 - 0.386i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.58 - 17.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.32024159596013908190507532693, −9.697034736137772112071841234169, −9.403104311317226886433194283441, −8.589077984676829376281879670624, −7.70729333693645860394539639219, −6.56654010076359122048979950710, −5.19604389848409196627361270052, −4.22269319550406653378471384054, −3.13012913885275154650534079417, −0.997235919315350598201205147787, 0.875079146648410727803461203673, 3.16006823056623655330001510809, 3.59986827985936951430386469100, 5.81310935457637412725545553537, 6.67115100158256351437013970428, 7.57226702588374944727871051564, 7.81684865584065223814089624000, 9.236292830351260579357928126034, 10.30865727872403957807190727804, 10.89659726253768602738649150382

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