Properties

Label 2-475-19.18-c2-0-19
Degree $2$
Conductor $475$
Sign $-0.836 - 0.547i$
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  + 1.47i·2-s + 3.55i·3-s + 1.82·4-s − 5.24·6-s + 7.05·7-s + 8.58i·8-s − 3.65·9-s + 2.24·11-s + 6.50i·12-s + 6.86i·13-s + 10.4i·14-s − 5.34·16-s + 11.1·17-s − 5.38i·18-s + (−15.8 − 10.4i)19-s + ⋯
L(s)  = 1  + 0.736i·2-s + 1.18i·3-s + 0.457·4-s − 0.873·6-s + 1.00·7-s + 1.07i·8-s − 0.406·9-s + 0.203·11-s + 0.542i·12-s + 0.527i·13-s + 0.743i·14-s − 0.333·16-s + 0.658·17-s − 0.299i·18-s + (−0.836 − 0.547i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.836 - 0.547i$
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: \((2,\ 475,\ (\ :1),\ -0.836 - 0.547i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.354772252\)
\(L(\frac12)\) \(\approx\) \(2.354772252\)
\(L(2)\) 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 \( 1 + (15.8 + 10.4i)T \)
good2 \( 1 - 1.47iT - 4T^{2} \)
3 \( 1 - 3.55iT - 9T^{2} \)
7 \( 1 - 7.05T + 49T^{2} \)
11 \( 1 - 2.24T + 121T^{2} \)
13 \( 1 - 6.86iT - 169T^{2} \)
17 \( 1 - 11.1T + 289T^{2} \)
23 \( 1 - 21.1T + 529T^{2} \)
29 \( 1 - 45.9iT - 841T^{2} \)
31 \( 1 + 14.7iT - 961T^{2} \)
37 \( 1 + 25.0iT - 1.36e3T^{2} \)
41 \( 1 + 64.9iT - 1.68e3T^{2} \)
43 \( 1 + 44.0T + 1.84e3T^{2} \)
47 \( 1 + 34.0T + 2.20e3T^{2} \)
53 \( 1 + 20.8iT - 2.80e3T^{2} \)
59 \( 1 + 69.2iT - 3.48e3T^{2} \)
61 \( 1 - 23.4T + 3.72e3T^{2} \)
67 \( 1 + 79.3iT - 4.48e3T^{2} \)
71 \( 1 + 26.8iT - 5.04e3T^{2} \)
73 \( 1 + 99.3T + 5.32e3T^{2} \)
79 \( 1 - 109. iT - 6.24e3T^{2} \)
83 \( 1 - 124.T + 6.88e3T^{2} \)
89 \( 1 + 85.7iT - 7.92e3T^{2} \)
97 \( 1 + 35.5iT - 9.40e3T^{2} \)
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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

−11.01260628113130890992174345271, −10.43486304221770451293044122401, −9.201429500572353705876376619192, −8.521313273071369178528947254784, −7.44939142933781364290082294307, −6.59910857796619143252167224542, −5.28460906328802391605889252251, −4.75712496902479508528213899660, −3.46378074741973852633848489422, −1.85850727841953326474215101378, 1.04534661875779085101655170841, 1.85949898081057164523163992167, 3.02724664550269427629121599687, 4.48482078704781317849860303796, 5.94197236225635084579940394731, 6.80502708340035394162200628155, 7.73695714378427488775239330088, 8.339651617677013124892709128072, 9.828288725764655794904223519651, 10.60378011115524084409274764381

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