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Properties

Label 2-476-119.10-c1-0-8
Degree $2$
Conductor $476$
Sign $-0.221 + 0.975i$
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  + (−0.951 − 1.92i)3-s + (3.19 − 2.79i)5-s + (1.81 − 1.92i)7-s + (−0.991 + 1.29i)9-s + (1.13 − 0.0742i)11-s + (3.38 + 3.38i)13-s + (−8.43 − 3.49i)15-s + (−1.61 + 3.79i)17-s + (0.342 + 2.59i)19-s + (−5.44 − 1.65i)21-s + (−1.44 − 0.714i)23-s + (1.69 − 12.8i)25-s + (−2.89 − 0.575i)27-s + (1.39 + 7.02i)29-s + (−2.18 + 1.07i)31-s + ⋯
L(s)  = 1  + (−0.549 − 1.11i)3-s + (1.42 − 1.25i)5-s + (0.684 − 0.728i)7-s + (−0.330 + 0.430i)9-s + (0.341 − 0.0223i)11-s + (0.937 + 0.937i)13-s + (−2.17 − 0.902i)15-s + (−0.391 + 0.920i)17-s + (0.0785 + 0.596i)19-s + (−1.18 − 0.362i)21-s + (−0.302 − 0.148i)23-s + (0.339 − 2.57i)25-s + (−0.557 − 0.110i)27-s + (0.259 + 1.30i)29-s + (−0.392 + 0.193i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $-0.221 + 0.975i$
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: \((2,\ 476,\ (\ :1/2),\ -0.221 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03387 - 1.29475i\)
\(L(\frac12)\) \(\approx\) \(1.03387 - 1.29475i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 + 1.92i)T \)
17 \( 1 + (1.61 - 3.79i)T \)
good3 \( 1 + (0.951 + 1.92i)T + (-1.82 + 2.38i)T^{2} \)
5 \( 1 + (-3.19 + 2.79i)T + (0.652 - 4.95i)T^{2} \)
11 \( 1 + (-1.13 + 0.0742i)T + (10.9 - 1.43i)T^{2} \)
13 \( 1 + (-3.38 - 3.38i)T + 13iT^{2} \)
19 \( 1 + (-0.342 - 2.59i)T + (-18.3 + 4.91i)T^{2} \)
23 \( 1 + (1.44 + 0.714i)T + (14.0 + 18.2i)T^{2} \)
29 \( 1 + (-1.39 - 7.02i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (2.18 - 1.07i)T + (18.8 - 24.5i)T^{2} \)
37 \( 1 + (0.312 - 4.76i)T + (-36.6 - 4.82i)T^{2} \)
41 \( 1 + (1.07 - 5.39i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (2.41 + 5.83i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (1.67 - 6.23i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.29 + 8.20i)T + (-13.7 + 51.1i)T^{2} \)
59 \( 1 + (6.62 + 0.872i)T + (56.9 + 15.2i)T^{2} \)
61 \( 1 + (-3.08 - 9.08i)T + (-48.3 + 37.1i)T^{2} \)
67 \( 1 + (4.42 + 2.55i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.82 - 1.89i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (-1.40 - 0.476i)T + (57.9 + 44.4i)T^{2} \)
79 \( 1 + (5.29 - 10.7i)T + (-48.0 - 62.6i)T^{2} \)
83 \( 1 + (5.81 - 14.0i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (1.50 + 0.403i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-15.8 + 3.15i)T + (89.6 - 37.1i)T^{2} \)
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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

−10.83534534848884409857042871445, −9.844550757755530599124197854166, −8.817547348276293128728097699391, −8.146937502095732262371991065826, −6.75929950331032039693160320350, −6.20610440821376850263558685083, −5.22485734034695650459207774104, −4.13339134494885328303927366636, −1.63005566369249592255550265378, −1.39585857736458637107098392478, 2.10213942323596683243730825282, 3.27460062719398352302284290135, 4.76487929964109958125855460275, 5.69557858812245686069530625679, 6.22365689850062963081340331035, 7.53042991190406721863168586625, 8.968336404646523615012669332295, 9.638926467916073160323376236064, 10.44708672403342955259373694063, 11.04937692913201771272948031327

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