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Properties

Label 2-476-7.3-c2-0-18
Degree $2$
Conductor $476$
Sign $-0.982 - 0.186i$
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  + (−4.95 − 2.85i)3-s + (2.87 − 1.65i)5-s + (−4.68 − 5.20i)7-s + (11.8 + 20.5i)9-s + (8.80 − 15.2i)11-s − 23.2i·13-s − 18.9·15-s + (3.57 + 2.06i)17-s + (17.8 − 10.3i)19-s + (8.29 + 39.1i)21-s + (−9.88 − 17.1i)23-s + (−6.98 + 12.1i)25-s − 83.8i·27-s − 24.3·29-s + (30.1 + 17.4i)31-s + ⋯
L(s)  = 1  + (−1.65 − 0.952i)3-s + (0.574 − 0.331i)5-s + (−0.668 − 0.743i)7-s + (1.31 + 2.27i)9-s + (0.800 − 1.38i)11-s − 1.78i·13-s − 1.26·15-s + (0.210 + 0.121i)17-s + (0.941 − 0.543i)19-s + (0.395 + 1.86i)21-s + (−0.429 − 0.744i)23-s + (−0.279 + 0.484i)25-s − 3.10i·27-s − 0.840·29-s + (0.973 + 0.562i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $-0.982 - 0.186i$
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: \((2,\ 476,\ (\ :1),\ -0.982 - 0.186i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7719389906\)
\(L(\frac12)\) \(\approx\) \(0.7719389906\)
\(L(2)\) 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 \( 1 + (4.68 + 5.20i)T \)
17 \( 1 + (-3.57 - 2.06i)T \)
good3 \( 1 + (4.95 + 2.85i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-2.87 + 1.65i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-8.80 + 15.2i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 23.2iT - 169T^{2} \)
19 \( 1 + (-17.8 + 10.3i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (9.88 + 17.1i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 24.3T + 841T^{2} \)
31 \( 1 + (-30.1 - 17.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-7.44 - 12.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 10.4iT - 1.68e3T^{2} \)
43 \( 1 + 42.9T + 1.84e3T^{2} \)
47 \( 1 + (25.1 - 14.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (1.00 - 1.73i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (65.9 + 38.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-73.0 + 42.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (48.5 - 84.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 41.9T + 5.04e3T^{2} \)
73 \( 1 + (59.8 + 34.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (6.84 + 11.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 37.5iT - 6.88e3T^{2} \)
89 \( 1 + (-24.1 + 13.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 72.2iT - 9.40e3T^{2} \)
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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

−10.50122156315459872619199202828, −9.765228462499828559728404089749, −8.272552727574951962663905753713, −7.33951385865304265570516858645, −6.32543344640271263174946401480, −5.83995344789617502844348877323, −4.95152468214723317918318410770, −3.26138177245709396519035820663, −1.21928182669448937056170562555, −0.45022135184254710281957180516, 1.78050654086616670268734360414, 3.77968883318860041519554288299, 4.66290942366860265655747543309, 5.73718856507336612689075494199, 6.39864183087032735927582278784, 7.11937874847586867904098550110, 9.240681941911536654364029784187, 9.684559674776275298220878970209, 10.12998494797433293035481836713, 11.57516683968085689809031778553

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