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Properties

Label 2-476-7.5-c2-0-21
Degree $2$
Conductor $476$
Sign $-0.302 + 0.953i$
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  + (4.85 − 2.80i)3-s + (−0.759 − 0.438i)5-s + (−2.35 − 6.59i)7-s + (11.2 − 19.4i)9-s + (−3.38 − 5.85i)11-s + 4.36i·13-s − 4.91·15-s + (−3.57 + 2.06i)17-s + (−5.04 − 2.90i)19-s + (−29.8 − 25.4i)21-s + (−21.3 + 36.8i)23-s + (−12.1 − 20.9i)25-s − 75.1i·27-s + 53.4·29-s + (46.0 − 26.5i)31-s + ⋯
L(s)  = 1  + (1.61 − 0.934i)3-s + (−0.151 − 0.0876i)5-s + (−0.336 − 0.941i)7-s + (1.24 − 2.15i)9-s + (−0.307 − 0.532i)11-s + 0.335i·13-s − 0.327·15-s + (−0.210 + 0.121i)17-s + (−0.265 − 0.153i)19-s + (−1.42 − 1.20i)21-s + (−0.926 + 1.60i)23-s + (−0.484 − 0.839i)25-s − 2.78i·27-s + 1.84·29-s + (1.48 − 0.857i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $-0.302 + 0.953i$
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: \((2,\ 476,\ (\ :1),\ -0.302 + 0.953i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.673835132\)
\(L(\frac12)\) \(\approx\) \(2.673835132\)
\(L(2)\) 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 \( 1 + (2.35 + 6.59i)T \)
17 \( 1 + (3.57 - 2.06i)T \)
good3 \( 1 + (-4.85 + 2.80i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (0.759 + 0.438i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (3.38 + 5.85i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 4.36iT - 169T^{2} \)
19 \( 1 + (5.04 + 2.90i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (21.3 - 36.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 53.4T + 841T^{2} \)
31 \( 1 + (-46.0 + 26.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (22.0 - 38.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 26.0iT - 1.68e3T^{2} \)
43 \( 1 - 19.9T + 1.84e3T^{2} \)
47 \( 1 + (-43.7 - 25.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (41.3 + 71.7i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-18.4 + 10.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-37.2 - 21.4i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-26.9 - 46.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 106.T + 5.04e3T^{2} \)
73 \( 1 + (41.2 - 23.8i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-39.8 + 69.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 106. iT - 6.88e3T^{2} \)
89 \( 1 + (-10.9 - 6.32i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 119. iT - 9.40e3T^{2} \)
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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

−10.20207893888197002331240870997, −9.574806492651352999272560800786, −8.369447275399289148773408955967, −8.024578889749995178277369645072, −7.01871994523277905654176071693, −6.26235377030097343122174738798, −4.29965092284750808111744639170, −3.41621044713176340525349520425, −2.31924125173876173764987459364, −0.916419128045697837214452620578, 2.26939146755975328777707727414, 2.95041243323269638134920735403, 4.14935472552793330109333554753, 5.04368511176363404871010969725, 6.52148128725747430004454326240, 7.85287389357827721591677774213, 8.477017647869568034009263711141, 9.183658510293912296577916939531, 10.08508817606221741471858712646, 10.59154085910728672783976581270

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