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Properties

Label 2-476-7.3-c2-0-16
Degree $2$
Conductor $476$
Sign $0.930 - 0.366i$
Analytic cond. $12.9700$
Root an. cond. $3.60139$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.90 + 2.25i)3-s + (2.01 − 1.16i)5-s + (6.73 + 1.90i)7-s + (5.69 + 9.85i)9-s + (7.01 − 12.1i)11-s − 15.2i·13-s + 10.5·15-s + (3.57 + 2.06i)17-s + (0.975 − 0.563i)19-s + (22.0 + 22.6i)21-s + (7.99 + 13.8i)23-s + (−9.78 + 16.9i)25-s + 10.7i·27-s − 41.4·29-s + (−1.11 − 0.641i)31-s + ⋯
L(s)  = 1  + (1.30 + 0.752i)3-s + (0.403 − 0.233i)5-s + (0.962 + 0.272i)7-s + (0.632 + 1.09i)9-s + (0.638 − 1.10i)11-s − 1.17i·13-s + 0.701·15-s + (0.210 + 0.121i)17-s + (0.0513 − 0.0296i)19-s + (1.04 + 1.07i)21-s + (0.347 + 0.602i)23-s + (−0.391 + 0.677i)25-s + 0.398i·27-s − 1.42·29-s + (−0.0358 − 0.0207i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.930 - 0.366i$
Analytic conductor: \(12.9700\)
Root analytic conductor: \(3.60139\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1),\ 0.930 - 0.366i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.254692440\)
\(L(\frac12)\) \(\approx\) \(3.254692440\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-6.73 - 1.90i)T \)
17 \( 1 + (-3.57 - 2.06i)T \)
good3 \( 1 + (-3.90 - 2.25i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-2.01 + 1.16i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-7.01 + 12.1i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 15.2iT - 169T^{2} \)
19 \( 1 + (-0.975 + 0.563i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-7.99 - 13.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 41.4T + 841T^{2} \)
31 \( 1 + (1.11 + 0.641i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-3.69 - 6.39i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 31.6iT - 1.68e3T^{2} \)
43 \( 1 + 6.24T + 1.84e3T^{2} \)
47 \( 1 + (33.9 - 19.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (7.58 - 13.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (77.9 + 45.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (84.5 - 48.7i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (9.99 - 17.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 68.9T + 5.04e3T^{2} \)
73 \( 1 + (-77.6 - 44.8i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-18.7 - 32.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 46.2iT - 6.88e3T^{2} \)
89 \( 1 + (-37.6 + 21.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 55.2iT - 9.40e3T^{2} \)
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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

−10.80466123911908048633753099022, −9.651282060509890727166090482762, −9.103011322223738663499763348980, −8.244643868911987487377939000899, −7.67530615607877203456405031310, −5.93349318702584782885768587498, −5.06813423820234224346920866134, −3.78674285028583271138753769265, −2.95064046261111769536707762187, −1.50512283744602978217852520351, 1.64152614634399716719965535301, 2.20797485118667091789944080489, 3.75955991659978048277944333906, 4.80282165798390794602827578593, 6.43800175819397465031081335049, 7.26306842487733426811779330044, 7.914752577840326378897995697841, 8.987117275085521890739466425004, 9.513229033670948570816763458546, 10.70146530472502913448497512419

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