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Properties

Label 2-476-119.10-c1-0-11
Degree $2$
Conductor $476$
Sign $-0.926 + 0.375i$
Analytic cond. $3.80087$
Root an. cond. $1.94958$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.926 − 1.87i)3-s + (1.35 − 1.18i)5-s + (−2.62 + 0.296i)7-s + (−0.844 + 1.10i)9-s + (1.97 − 0.129i)11-s + (−4.57 − 4.57i)13-s + (−3.48 − 1.44i)15-s + (2.52 − 3.25i)17-s + (0.436 + 3.31i)19-s + (2.99 + 4.66i)21-s + (−6.66 − 3.28i)23-s + (−0.230 + 1.74i)25-s + (−3.31 − 0.659i)27-s + (1.43 + 7.20i)29-s + (−5.67 + 2.79i)31-s + ⋯
L(s)  = 1  + (−0.534 − 1.08i)3-s + (0.604 − 0.530i)5-s + (−0.993 + 0.112i)7-s + (−0.281 + 0.366i)9-s + (0.595 − 0.0390i)11-s + (−1.26 − 1.26i)13-s + (−0.898 − 0.372i)15-s + (0.613 − 0.789i)17-s + (0.100 + 0.761i)19-s + (0.653 + 1.01i)21-s + (−1.39 − 0.685i)23-s + (−0.0460 + 0.349i)25-s + (−0.637 − 0.126i)27-s + (0.266 + 1.33i)29-s + (−1.01 + 0.502i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $-0.926 + 0.375i$
Analytic conductor: \(3.80087\)
Root analytic conductor: \(1.94958\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1/2),\ -0.926 + 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.164391 - 0.842703i\)
\(L(\frac12)\) \(\approx\) \(0.164391 - 0.842703i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 - 0.296i)T \)
17 \( 1 + (-2.52 + 3.25i)T \)
good3 \( 1 + (0.926 + 1.87i)T + (-1.82 + 2.38i)T^{2} \)
5 \( 1 + (-1.35 + 1.18i)T + (0.652 - 4.95i)T^{2} \)
11 \( 1 + (-1.97 + 0.129i)T + (10.9 - 1.43i)T^{2} \)
13 \( 1 + (4.57 + 4.57i)T + 13iT^{2} \)
19 \( 1 + (-0.436 - 3.31i)T + (-18.3 + 4.91i)T^{2} \)
23 \( 1 + (6.66 + 3.28i)T + (14.0 + 18.2i)T^{2} \)
29 \( 1 + (-1.43 - 7.20i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (5.67 - 2.79i)T + (18.8 - 24.5i)T^{2} \)
37 \( 1 + (-0.358 + 5.47i)T + (-36.6 - 4.82i)T^{2} \)
41 \( 1 + (-1.73 + 8.71i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (0.466 + 1.12i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (-2.08 + 7.78i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.61 - 4.70i)T + (-13.7 + 51.1i)T^{2} \)
59 \( 1 + (-5.78 - 0.761i)T + (56.9 + 15.2i)T^{2} \)
61 \( 1 + (2.37 + 7.00i)T + (-48.3 + 37.1i)T^{2} \)
67 \( 1 + (-1.63 - 0.941i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.85 + 1.24i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (-6.44 - 2.18i)T + (57.9 + 44.4i)T^{2} \)
79 \( 1 + (-7.17 + 14.5i)T + (-48.0 - 62.6i)T^{2} \)
83 \( 1 + (4.78 - 11.5i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (1.46 + 0.392i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.07 + 1.80i)T + (89.6 - 37.1i)T^{2} \)
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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

−10.46463979817130922114969686721, −9.769992978385343485102469500588, −8.900085598777699905906442032019, −7.57846241228548815351551662446, −6.96418717209031659080144531809, −5.84417130206958832372110953316, −5.33937472343293960345085150251, −3.52914784981291340360597620966, −2.06086538233089347963581045099, −0.53674544658568788187425759886, 2.28294586800471200319405321656, 3.76639763982778711269907465784, 4.58457716924482936754591548863, 5.89070067023263003676833560427, 6.51096480027322628112717211025, 7.63159296027225095804171876231, 9.240008407529771700186084075020, 9.879655762571071779453433446659, 10.10798325477453136985141257732, 11.36621743040786643158125938830

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