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Properties

Label 2-476-7.3-c2-0-21
Degree $2$
Conductor $476$
Sign $-0.992 - 0.123i$
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  + (−0.565 − 0.326i)3-s + (1.39 − 0.802i)5-s + (−5.00 − 4.89i)7-s + (−4.28 − 7.42i)9-s + (4.70 − 8.15i)11-s + 22.3i·13-s − 1.04·15-s + (3.57 + 2.06i)17-s + (−27.5 + 15.8i)19-s + (1.23 + 4.39i)21-s + (−11.5 − 19.9i)23-s + (−11.2 + 19.4i)25-s + 11.4i·27-s − 29.1·29-s + (12.0 + 6.95i)31-s + ⋯
L(s)  = 1  + (−0.188 − 0.108i)3-s + (0.278 − 0.160i)5-s + (−0.714 − 0.699i)7-s + (−0.476 − 0.825i)9-s + (0.427 − 0.740i)11-s + 1.71i·13-s − 0.0698·15-s + (0.210 + 0.121i)17-s + (−1.44 + 0.836i)19-s + (0.0585 + 0.209i)21-s + (−0.500 − 0.866i)23-s + (−0.448 + 0.776i)25-s + 0.424i·27-s − 1.00·29-s + (0.388 + 0.224i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $-0.992 - 0.123i$
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: \((2,\ 476,\ (\ :1),\ -0.992 - 0.123i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2441832043\)
\(L(\frac12)\) \(\approx\) \(0.2441832043\)
\(L(2)\) 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 \( 1 + (5.00 + 4.89i)T \)
17 \( 1 + (-3.57 - 2.06i)T \)
good3 \( 1 + (0.565 + 0.326i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-1.39 + 0.802i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-4.70 + 8.15i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 22.3iT - 169T^{2} \)
19 \( 1 + (27.5 - 15.8i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 29.1T + 841T^{2} \)
31 \( 1 + (-12.0 - 6.95i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (24.4 + 42.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 44.6iT - 1.68e3T^{2} \)
43 \( 1 + 1.15T + 1.84e3T^{2} \)
47 \( 1 + (-18.6 + 10.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (52.4 - 90.7i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (77.8 + 44.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (60.6 - 35.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-35.2 + 61.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 22.3T + 5.04e3T^{2} \)
73 \( 1 + (30.2 + 17.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (10.6 + 18.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 12.2iT - 6.88e3T^{2} \)
89 \( 1 + (-82.5 + 47.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 95.0iT - 9.40e3T^{2} \)
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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

−10.39489211248343664014491754856, −9.222801273694252153065640328391, −8.872537007195319834446361994785, −7.43197978068419683500034843908, −6.36350253669547517395729023659, −6.00718113502378154439648821386, −4.26283463648439159882842054846, −3.52402875885706117122914368818, −1.76677201532505420392651641343, −0.095116659255417409158162523054, 2.14723916298590081092271765208, 3.18098177060707982675182907365, 4.71113185759247157318660958101, 5.70590084880536132863786996770, 6.43988076445677861541295214224, 7.72532652020849936943526409961, 8.530811691307129807123652446515, 9.664705965371374512192173622885, 10.26598914531740662700118134074, 11.19431747868277042391809765266

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