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Properties

Label 2-476-7.5-c2-0-12
Degree $2$
Conductor $476$
Sign $0.941 + 0.338i$
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  + (−4.26 + 2.46i)3-s + (3.82 + 2.20i)5-s + (6.67 − 2.10i)7-s + (7.61 − 13.1i)9-s + (−7.01 − 12.1i)11-s + 7.16i·13-s − 21.7·15-s + (3.57 − 2.06i)17-s + (−28.5 − 16.5i)19-s + (−23.2 + 25.4i)21-s + (19.5 − 33.8i)23-s + (−2.75 − 4.76i)25-s + 30.6i·27-s + 18.9·29-s + (30.1 − 17.3i)31-s + ⋯
L(s)  = 1  + (−1.42 + 0.820i)3-s + (0.764 + 0.441i)5-s + (0.953 − 0.300i)7-s + (0.846 − 1.46i)9-s + (−0.637 − 1.10i)11-s + 0.551i·13-s − 1.44·15-s + (0.210 − 0.121i)17-s + (−1.50 − 0.868i)19-s + (−1.10 + 1.20i)21-s + (0.849 − 1.47i)23-s + (−0.110 − 0.190i)25-s + 1.13i·27-s + 0.654·29-s + (0.971 − 0.560i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.338i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.941 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.941 + 0.338i$
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: \((2,\ 476,\ (\ :1),\ 0.941 + 0.338i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.151177606\)
\(L(\frac12)\) \(\approx\) \(1.151177606\)
\(L(2)\) 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 \( 1 + (-6.67 + 2.10i)T \)
17 \( 1 + (-3.57 + 2.06i)T \)
good3 \( 1 + (4.26 - 2.46i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3.82 - 2.20i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (7.01 + 12.1i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 7.16iT - 169T^{2} \)
19 \( 1 + (28.5 + 16.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-19.5 + 33.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 18.9T + 841T^{2} \)
31 \( 1 + (-30.1 + 17.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (8.39 - 14.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 51.2iT - 1.68e3T^{2} \)
43 \( 1 - 68.4T + 1.84e3T^{2} \)
47 \( 1 + (-40.5 - 23.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (30.9 + 53.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (6.28 - 3.62i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (33.4 + 19.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (32.3 + 56.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 67.4T + 5.04e3T^{2} \)
73 \( 1 + (-87.0 + 50.2i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (0.257 - 0.445i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 12.1iT - 6.88e3T^{2} \)
89 \( 1 + (-128. - 74.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 7.13iT - 9.40e3T^{2} \)
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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

−10.75917106999918867917073352033, −10.32149905567302772313025413682, −9.127053942101608507189294473190, −8.103071971581046714190578613244, −6.54792092435343013977760669075, −6.11992434055925668026274561933, −4.93334531204915119266421535415, −4.38867864933626946901080513047, −2.56776446498283232623782124538, −0.64163205020694821626683840096, 1.22795609497890736680721602741, 2.15008795916766258571146837439, 4.53594418679314454015206552683, 5.42881486873813766988260705249, 5.88905232135908432632583259131, 7.14319785455948030853063095057, 7.87075997474572791298179742778, 9.049876820810785934984743562361, 10.35183548339497276659217014090, 10.77105117717420558388000323186

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