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Properties

Label 2-476-7.4-c1-0-1
Degree $2$
Conductor $476$
Sign $-0.605 - 0.795i$
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  + (−0.5 + 0.866i)3-s + (2 + 3.46i)5-s + (−2 + 1.73i)7-s + (1 + 1.73i)9-s + (1.5 − 2.59i)11-s − 5·13-s − 3.99·15-s + (0.5 − 0.866i)17-s + (−3 − 5.19i)19-s + (−0.499 − 2.59i)21-s + (2 + 3.46i)23-s + (−5.49 + 9.52i)25-s − 5·27-s + 8·29-s + (1.5 + 2.59i)33-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.894 + 1.54i)5-s + (−0.755 + 0.654i)7-s + (0.333 + 0.577i)9-s + (0.452 − 0.783i)11-s − 1.38·13-s − 1.03·15-s + (0.121 − 0.210i)17-s + (−0.688 − 1.19i)19-s + (−0.109 − 0.566i)21-s + (0.417 + 0.722i)23-s + (−1.09 + 1.90i)25-s − 0.962·27-s + 1.48·29-s + (0.261 + 0.452i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(3.80087\)
Root analytic conductor: \(1.94958\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.547015 + 1.10346i\)
\(L(\frac12)\) \(\approx\) \(0.547015 + 1.10346i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7T + 71T^{2} \)
73 \( 1 + (-4 + 6.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.5 + 12.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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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

−11.14016967911275198332198693851, −10.30833381592883759392033763398, −9.804646222048664801357594556682, −8.939380906251546439201050092857, −7.38835861956298908950794173935, −6.61798013325581887009937250510, −5.82696301981483944496796448661, −4.74676173703891443300394719050, −3.10334143367717061002397666597, −2.41017011917080799956297620662, 0.77736879848003819330441892225, 2.06167778531779518806792174935, 4.05756286642614708584885173730, 4.92772260857320028913971866498, 6.12057786449719530044730100420, 6.83022545110749464947171567896, 7.939229705611444732851104076085, 9.135565609968013842425093580532, 9.739531373154519506040565257706, 10.35486219380780348092691761783

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