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Properties

Label 2-476-7.5-c2-0-11
Degree $2$
Conductor $476$
Sign $0.937 + 0.348i$
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  + (−0.354 + 0.204i)3-s + (−3.41 − 1.96i)5-s + (6.69 − 2.02i)7-s + (−4.41 + 7.64i)9-s + (3.71 + 6.43i)11-s − 13.2i·13-s + 1.61·15-s + (3.57 − 2.06i)17-s + (14.1 + 8.18i)19-s + (−1.96 + 2.09i)21-s + (0.578 − 1.00i)23-s + (−4.73 − 8.20i)25-s − 7.30i·27-s + 32.1·29-s + (49.1 − 28.3i)31-s + ⋯
L(s)  = 1  + (−0.118 + 0.0683i)3-s + (−0.682 − 0.393i)5-s + (0.957 − 0.289i)7-s + (−0.490 + 0.849i)9-s + (0.338 + 0.585i)11-s − 1.02i·13-s + 0.107·15-s + (0.210 − 0.121i)17-s + (0.746 + 0.430i)19-s + (−0.0934 + 0.0996i)21-s + (0.0251 − 0.0436i)23-s + (−0.189 − 0.328i)25-s − 0.270i·27-s + 1.10·29-s + (1.58 − 0.915i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.937 + 0.348i$
Analytic conductor: \(12.9700\)
Root analytic conductor: \(3.60139\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1),\ 0.937 + 0.348i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.614481117\)
\(L(\frac12)\) \(\approx\) \(1.614481117\)
\(L(2)\) 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 \( 1 + (-6.69 + 2.02i)T \)
17 \( 1 + (-3.57 + 2.06i)T \)
good3 \( 1 + (0.354 - 0.204i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (3.41 + 1.96i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.71 - 6.43i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 13.2iT - 169T^{2} \)
19 \( 1 + (-14.1 - 8.18i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-0.578 + 1.00i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 32.1T + 841T^{2} \)
31 \( 1 + (-49.1 + 28.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-5.88 + 10.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 41.3iT - 1.68e3T^{2} \)
43 \( 1 - 3.75T + 1.84e3T^{2} \)
47 \( 1 + (-37.8 - 21.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-39.0 - 67.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-22.8 + 13.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-82.7 - 47.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (29.2 + 50.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 74.1T + 5.04e3T^{2} \)
73 \( 1 + (85.8 - 49.5i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-65.4 + 113. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 36.5iT - 6.88e3T^{2} \)
89 \( 1 + (-96.0 - 55.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 33.5iT - 9.40e3T^{2} \)
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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

−10.69806238566651715590152444370, −10.08702481984721977202489385198, −8.708556867628125892156395624006, −7.945847099449446687704618267937, −7.44476890461465183745349844853, −5.86678665096956823023891621229, −4.91118167009019290281817293197, −4.11958793289546090259809691799, −2.55548196157111577226112800515, −0.895904530642741467981641788767, 1.09106585051869126420986225372, 2.85982588506374249241534592247, 3.97657927888019471861577683969, 5.10568209465054628765372914800, 6.30051897926430722355921639028, 7.11553101835445992038288629771, 8.270943660287822627003893311416, 8.852900669577361934766251666105, 9.963525570152703578426621942151, 11.19777744595105699117341309185

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