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Properties

Label 2-475-95.94-c2-0-29
Degree $2$
Conductor $475$
Sign $0.201 + 0.979i$
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  − 3.82·2-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.201 + 0.979i$
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: \((2,\ 475,\ (\ :1),\ 0.201 + 0.979i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3327140887\)
\(L(\frac12)\) \(\approx\) \(0.3327140887\)
\(L(2)\) 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 \( 1 + 14.5T + 9.40e3T^{2} \)
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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

−10.41594067709951997825583572690, −9.444385966486817867878909930324, −8.716695167273225580609842724973, −8.432695054001437909611204077963, −7.09366539317240728266715868471, −6.34276984341845701677547938262, −5.23527584879103905359003071184, −2.81922555567464301685230164929, −2.20206837593732917174805362970, −0.29190741271606642146363749789, 1.02746680235961156873657433671, 2.43138238451387993990849110078, 3.87251975301379491561958233426, 5.82783797443290543274056705445, 6.78898337926662561291806709537, 7.73233132994210627059925720966, 8.130355152583197647758404466896, 9.470929967253453908327917106329, 9.798772225471806042594498258016, 10.83825187718479690409630986685

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