↑

Properties

Label 2-475-95.94-c2-0-40
Degree $2$
Conductor $475$
Sign $0.904 - 0.426i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.82·2-s − 0.102·3-s + 10.6·4-s − 0.390·6-s + 9.96i·7-s + 25.5·8-s − 8.98·9-s + 5.29·11-s − 1.08·12-s + 13.5·13-s + 38.1i·14-s + 55.1·16-s − 5.23i·17-s − 34.4·18-s + (−11.7 + 14.9i)19-s + ⋯
L(s)  = 1  + 1.91·2-s − 0.0340·3-s + 2.66·4-s − 0.0651·6-s + 1.42i·7-s + 3.19·8-s − 0.998·9-s + 0.481·11-s − 0.0907·12-s + 1.04·13-s + 2.72i·14-s + 3.44·16-s − 0.307i·17-s − 1.91·18-s + (−0.618 + 0.785i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.904 - 0.426i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (474, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ 0.904 - 0.426i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.573803304\)
\(L(\frac12)\) \(\approx\) \(5.573803304\)
\(L(2)\) 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 \( 1 - 3.82T + 4T^{2} \)
3 \( 1 + 0.102T + 9T^{2} \)
7 \( 1 - 9.96iT - 49T^{2} \)
11 \( 1 - 5.29T + 121T^{2} \)
13 \( 1 - 13.5T + 169T^{2} \)
17 \( 1 + 5.23iT - 289T^{2} \)
23 \( 1 + 37.5iT - 529T^{2} \)
29 \( 1 + 26.4iT - 841T^{2} \)
31 \( 1 + 29.3iT - 961T^{2} \)
37 \( 1 + 38.9T + 1.36e3T^{2} \)
41 \( 1 - 39.5iT - 1.68e3T^{2} \)
43 \( 1 - 30.3iT - 1.84e3T^{2} \)
47 \( 1 + 77.0iT - 2.20e3T^{2} \)
53 \( 1 + 41.0T + 2.80e3T^{2} \)
59 \( 1 - 50.0iT - 3.48e3T^{2} \)
61 \( 1 - 32.2T + 3.72e3T^{2} \)
67 \( 1 + 100.T + 4.48e3T^{2} \)
71 \( 1 + 90.8iT - 5.04e3T^{2} \)
73 \( 1 - 4.79iT - 5.32e3T^{2} \)
79 \( 1 - 128. iT - 6.24e3T^{2} \)
83 \( 1 + 100. iT - 6.88e3T^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 - 14.5T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.42733372361019243129342013712, −10.42112151136050007833235025769, −8.874036365645937000580926013924, −8.050951723690795496690108087802, −6.39437182963438409696884624848, −6.10787269727116114340840573818, −5.21134309893403320249061086065, −4.08988094522614175042472903585, −2.98228411255402759388285460209, −2.08009186040827903193581754206, 1.51731815365050931682463076559, 3.23926290441414621714162489188, 3.82093574373934520736544079030, 4.89625285275890080169606364873, 5.89060034503809704342108339040, 6.74106155730315462930885704182, 7.49821903051811261699574970664, 8.815937140300803094090700342394, 10.49277820548957686438044422588, 11.01232879985115280696025741355

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