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Properties

Label 2-475-19.6-c1-0-27
Degree $2$
Conductor $475$
Sign $-0.766 + 0.642i$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
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  + (1.93 − 1.62i)2-s + (−0.613 − 0.223i)3-s + (0.766 − 4.34i)4-s + (−1.55 + 0.565i)6-s + (0.766 − 1.32i)7-s + (−3.05 − 5.28i)8-s + (−1.97 − 1.65i)9-s + (0.592 + 1.02i)11-s + (−1.43 + 2.49i)12-s + (2.55 − 0.929i)13-s + (−0.673 − 3.82i)14-s + (−6.23 − 2.27i)16-s + (−2.97 + 2.49i)17-s − 6.51·18-s + (0.819 + 4.28i)19-s + ⋯
L(s)  = 1  + (1.37 − 1.15i)2-s + (−0.354 − 0.128i)3-s + (0.383 − 2.17i)4-s + (−0.634 + 0.230i)6-s + (0.289 − 0.501i)7-s + (−1.07 − 1.86i)8-s + (−0.657 − 0.551i)9-s + (0.178 + 0.309i)11-s + (−0.415 + 0.719i)12-s + (0.708 − 0.257i)13-s + (−0.180 − 1.02i)14-s + (−1.55 − 0.567i)16-s + (−0.720 + 0.604i)17-s − 1.53·18-s + (0.187 + 0.982i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.862894 - 2.37319i\)
\(L(\frac12)\) \(\approx\) \(0.862894 - 2.37319i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (-0.819 - 4.28i)T \)
good2 \( 1 + (-1.93 + 1.62i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (0.613 + 0.223i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (-0.766 + 1.32i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.592 - 1.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.55 + 0.929i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.97 - 2.49i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.879 + 4.98i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (3.56 + 2.99i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.91 + 3.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.10T + 37T^{2} \)
41 \( 1 + (-9.38 - 3.41i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.51 - 8.57i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.439 + 0.368i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (0.511 - 2.89i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (3.01 - 2.52i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.784 - 4.44i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-2.97 - 2.49i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.20 + 6.83i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-5.75 - 2.09i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (9.21 + 3.35i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-6.15 + 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.27 + 0.829i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (5.64 - 4.73i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.03357065766215319831113723699, −10.26955975000529733860214224135, −9.186436661356085920762611397049, −7.87900588217252802393483089894, −6.29142023149100229053133542592, −5.90355351330263881033801139196, −4.54562635963173749537950294424, −3.85456904695521325878241066139, −2.62479363674641076701708720178, −1.16435142744464778876853045418, 2.62829401108111140533247952958, 3.87508685354755030001343598465, 5.03185593589349821270454713070, 5.54801085961529795610278109702, 6.50390969424696666966809393736, 7.38634600913871358391576300717, 8.422782203248790499657401078884, 9.191468004613452480081903465922, 11.03296000387390402053197139607, 11.43683972975018690506398142682

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