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Properties

Label 2-476-7.5-c2-0-17
Degree $2$
Conductor $476$
Sign $0.856 + 0.515i$
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  + (2.95 − 1.70i)3-s + (6.58 + 3.80i)5-s + (6.95 − 0.760i)7-s + (1.33 − 2.31i)9-s + (−10.0 − 17.3i)11-s − 21.2i·13-s + 25.9·15-s + (3.57 − 2.06i)17-s + (9.70 + 5.60i)19-s + (19.2 − 14.1i)21-s + (−18.0 + 31.3i)23-s + (16.4 + 28.4i)25-s + 21.6i·27-s + 35.0·29-s + (−41.9 + 24.1i)31-s + ⋯
L(s)  = 1  + (0.986 − 0.569i)3-s + (1.31 + 0.760i)5-s + (0.994 − 0.108i)7-s + (0.148 − 0.257i)9-s + (−0.909 − 1.57i)11-s − 1.63i·13-s + 1.73·15-s + (0.210 − 0.121i)17-s + (0.510 + 0.294i)19-s + (0.918 − 0.673i)21-s + (−0.786 + 1.36i)23-s + (0.657 + 1.13i)25-s + 0.800i·27-s + 1.20·29-s + (−1.35 + 0.780i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.856 + 0.515i$
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: \((2,\ 476,\ (\ :1),\ 0.856 + 0.515i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.171173829\)
\(L(\frac12)\) \(\approx\) \(3.171173829\)
\(L(2)\) 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 \( 1 + (-6.95 + 0.760i)T \)
17 \( 1 + (-3.57 + 2.06i)T \)
good3 \( 1 + (-2.95 + 1.70i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-6.58 - 3.80i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (10.0 + 17.3i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 21.2iT - 169T^{2} \)
19 \( 1 + (-9.70 - 5.60i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (18.0 - 31.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 35.0T + 841T^{2} \)
31 \( 1 + (41.9 - 24.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-21.6 + 37.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 44.5iT - 1.68e3T^{2} \)
43 \( 1 - 24.5T + 1.84e3T^{2} \)
47 \( 1 + (5.60 + 3.23i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (16.5 + 28.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (40.6 - 23.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-5.98 - 3.45i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-43.5 - 75.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 100.T + 5.04e3T^{2} \)
73 \( 1 + (69.4 - 40.0i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (29.5 - 51.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 63.0iT - 6.88e3T^{2} \)
89 \( 1 + (121. + 70.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 21.0iT - 9.40e3T^{2} \)
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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

−10.64628079972367168422233811959, −9.921718278211688995721248719429, −8.692729011890228483752998038282, −7.982160916960167431810458301684, −7.37082610899851792243056727203, −5.74732514740699414449041984588, −5.48449254513354818894275603442, −3.25281994767331958231294189036, −2.63319715013839319685447710651, −1.36466251966445243896011144346, 1.79127575991503011191280523552, 2.42568555723034153996581001386, 4.39400108697783443708979070964, 4.80692159567235608691591225049, 6.08865390673524130275394733167, 7.40812225173667051842671399358, 8.445470731341235787736792285115, 9.181020128278544048915485487648, 9.747955097670387920749876972246, 10.51299692015437170120971287665

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