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Properties

Label 2-475-19.18-c2-0-26
Degree $2$
Conductor $475$
Sign $-0.618 - 0.785i$
Analytic cond. $12.9428$
Root an. cond. $3.59761$
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  + 3.82i·2-s + 0.102i·3-s − 10.6·4-s − 0.390·6-s + 9.96·7-s − 25.5i·8-s + 8.98·9-s + 5.29·11-s − 1.08i·12-s − 13.5i·13-s + 38.1i·14-s + 55.1·16-s − 5.23·17-s + 34.4i·18-s + (11.7 + 14.9i)19-s + ⋯
L(s)  = 1  + 1.91i·2-s + 0.0340i·3-s − 2.66·4-s − 0.0651·6-s + 1.42·7-s − 3.19i·8-s + 0.998·9-s + 0.481·11-s − 0.0907i·12-s − 1.04i·13-s + 2.72i·14-s + 3.44·16-s − 0.307·17-s + 1.91i·18-s + (0.618 + 0.785i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.618 - 0.785i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ -0.618 - 0.785i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.901998215\)
\(L(\frac12)\) \(\approx\) \(1.901998215\)
\(L(2)\) 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 \( 1 + (-11.7 - 14.9i)T \)
good2 \( 1 - 3.82iT - 4T^{2} \)
3 \( 1 - 0.102iT - 9T^{2} \)
7 \( 1 - 9.96T + 49T^{2} \)
11 \( 1 - 5.29T + 121T^{2} \)
13 \( 1 + 13.5iT - 169T^{2} \)
17 \( 1 + 5.23T + 289T^{2} \)
23 \( 1 - 37.5T + 529T^{2} \)
29 \( 1 + 26.4iT - 841T^{2} \)
31 \( 1 - 29.3iT - 961T^{2} \)
37 \( 1 + 38.9iT - 1.36e3T^{2} \)
41 \( 1 + 39.5iT - 1.68e3T^{2} \)
43 \( 1 + 30.3T + 1.84e3T^{2} \)
47 \( 1 + 77.0T + 2.20e3T^{2} \)
53 \( 1 - 41.0iT - 2.80e3T^{2} \)
59 \( 1 - 50.0iT - 3.48e3T^{2} \)
61 \( 1 - 32.2T + 3.72e3T^{2} \)
67 \( 1 + 100. iT - 4.48e3T^{2} \)
71 \( 1 - 90.8iT - 5.04e3T^{2} \)
73 \( 1 + 4.79T + 5.32e3T^{2} \)
79 \( 1 - 128. iT - 6.24e3T^{2} \)
83 \( 1 - 100.T + 6.88e3T^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 - 14.5iT - 9.40e3T^{2} \)
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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

−10.91580741555432466706905458573, −9.891552257163002237956566536779, −8.939097262099476557254458399877, −8.094981514027839887300515725930, −7.48113138701411832161557014707, −6.66914372195846271412455042467, −5.42846762548622119539807841958, −4.86434264466655282438248479760, −3.81666120506768265968484081880, −1.14228508347233088482074425667, 1.17525717350210098615795233537, 1.90433120272541963114814880198, 3.34214382933121809132986553127, 4.67074079102209842684586254391, 4.82812364981412034386908788803, 6.91195654433182999255851632190, 8.178097861668460384108954220173, 9.079107139538705679249663786907, 9.715134768087709999431048927895, 10.75993946734665547071874082127

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