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Properties

Label 2-475-19.18-c2-0-44
Degree $2$
Conductor $475$
Sign $0.422 + 0.906i$
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  + 0.382i·2-s − 3.15i·3-s + 3.85·4-s + 1.20·6-s + 4.26·7-s + 3.00i·8-s − 0.938·9-s − 9.61·11-s − 12.1i·12-s − 22.7i·13-s + 1.63i·14-s + 14.2·16-s + 25.9·17-s − 0.359i·18-s + (8.03 + 17.2i)19-s + ⋯
L(s)  = 1  + 0.191i·2-s − 1.05i·3-s + 0.963·4-s + 0.200·6-s + 0.609·7-s + 0.375i·8-s − 0.104·9-s − 0.874·11-s − 1.01i·12-s − 1.75i·13-s + 0.116i·14-s + 0.891·16-s + 1.52·17-s − 0.0199i·18-s + (0.422 + 0.906i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.422 + 0.906i$
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: \((2,\ 475,\ (\ :1),\ 0.422 + 0.906i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.375487959\)
\(L(\frac12)\) \(\approx\) \(2.375487959\)
\(L(2)\) 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 \( 1 + (-8.03 - 17.2i)T \)
good2 \( 1 - 0.382iT - 4T^{2} \)
3 \( 1 + 3.15iT - 9T^{2} \)
7 \( 1 - 4.26T + 49T^{2} \)
11 \( 1 + 9.61T + 121T^{2} \)
13 \( 1 + 22.7iT - 169T^{2} \)
17 \( 1 - 25.9T + 289T^{2} \)
23 \( 1 + 24.5T + 529T^{2} \)
29 \( 1 - 21.2iT - 841T^{2} \)
31 \( 1 - 6.67iT - 961T^{2} \)
37 \( 1 + 35.0iT - 1.36e3T^{2} \)
41 \( 1 + 72.2iT - 1.68e3T^{2} \)
43 \( 1 - 50.2T + 1.84e3T^{2} \)
47 \( 1 + 36.1T + 2.20e3T^{2} \)
53 \( 1 - 2.29iT - 2.80e3T^{2} \)
59 \( 1 - 19.0iT - 3.48e3T^{2} \)
61 \( 1 - 50.6T + 3.72e3T^{2} \)
67 \( 1 - 9.20iT - 4.48e3T^{2} \)
71 \( 1 - 116. iT - 5.04e3T^{2} \)
73 \( 1 + 49.2T + 5.32e3T^{2} \)
79 \( 1 + 87.5iT - 6.24e3T^{2} \)
83 \( 1 + 33.8T + 6.88e3T^{2} \)
89 \( 1 + 7.05iT - 7.92e3T^{2} \)
97 \( 1 + 16.0iT - 9.40e3T^{2} \)
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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

−10.51897465374985878477170792827, −10.15043296539432769254207005689, −8.236397730708522616369068667949, −7.74443188053334782183433088774, −7.27922537658740743868046141593, −5.85023154232092599645446004057, −5.44107317519887592165529116215, −3.40539771734937494842003930549, −2.22374595699555005591586123468, −1.05327680450258020517164190332, 1.62782633200982917378324098662, 2.95987253065211638552246734153, 4.17921564253165098499406296401, 5.08924034902404393792631227228, 6.26045876266139873725272478948, 7.37639770123488995142607646995, 8.187974868638990272508649844469, 9.642478163230072857013811928285, 9.947267284023194696978785094614, 11.07234628497242249514537604110

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