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Properties

Label 2-476-7.3-c2-0-8
Degree $2$
Conductor $476$
Sign $0.272 + 0.962i$
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  + (−3.19 − 1.84i)3-s + (−4.85 + 2.80i)5-s + (−2.56 + 6.51i)7-s + (2.29 + 3.97i)9-s + (−6.24 + 10.8i)11-s − 3.19i·13-s + 20.6·15-s + (3.57 + 2.06i)17-s + (25.6 − 14.7i)19-s + (20.1 − 16.0i)21-s + (−12.3 − 21.3i)23-s + (3.22 − 5.58i)25-s + 16.2i·27-s − 51.5·29-s + (9.63 + 5.56i)31-s + ⋯
L(s)  = 1  + (−1.06 − 0.614i)3-s + (−0.971 + 0.560i)5-s + (−0.365 + 0.930i)7-s + (0.255 + 0.441i)9-s + (−0.567 + 0.982i)11-s − 0.245i·13-s + 1.37·15-s + (0.210 + 0.121i)17-s + (1.34 − 0.778i)19-s + (0.961 − 0.765i)21-s + (−0.536 − 0.928i)23-s + (0.128 − 0.223i)25-s + 0.602i·27-s − 1.77·29-s + (0.310 + 0.179i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.272 + 0.962i$
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: \((2,\ 476,\ (\ :1),\ 0.272 + 0.962i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4855888171\)
\(L(\frac12)\) \(\approx\) \(0.4855888171\)
\(L(2)\) 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 \( 1 + (2.56 - 6.51i)T \)
17 \( 1 + (-3.57 - 2.06i)T \)
good3 \( 1 + (3.19 + 1.84i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (4.85 - 2.80i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (6.24 - 10.8i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 3.19iT - 169T^{2} \)
19 \( 1 + (-25.6 + 14.7i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (12.3 + 21.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 51.5T + 841T^{2} \)
31 \( 1 + (-9.63 - 5.56i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (6.17 + 10.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 75.7iT - 1.68e3T^{2} \)
43 \( 1 - 20.7T + 1.84e3T^{2} \)
47 \( 1 + (-78.5 + 45.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (8.43 - 14.6i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-21.9 - 12.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-15.1 + 8.75i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-2.14 + 3.71i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 57.6T + 5.04e3T^{2} \)
73 \( 1 + (-74.3 - 42.9i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (40.6 + 70.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 50.0iT - 6.88e3T^{2} \)
89 \( 1 + (93.0 - 53.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 47.2iT - 9.40e3T^{2} \)
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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

−10.91317950481457560915373158383, −9.854854586931445005378321570644, −8.770711793796506562003855464431, −7.42121962233552152824938856417, −7.13016144818487145067614791755, −5.86471170303640490610402505148, −5.17245344304143716736738471700, −3.66706418791315432974534029354, −2.32625643575922306404934483317, −0.32653389400626232804891800038, 0.851914609432812545733166011694, 3.42681926999655389417747562368, 4.21892923920748391642783469526, 5.29603366560564157018580138002, 6.05445094074241373219923747892, 7.51227570410868462518771940960, 8.030606474267471972725430762579, 9.453447246680914428641230873102, 10.18248748317830242094261489458, 11.23027437125287108066500819769

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