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Properties

Label 2-473-473.472-c1-0-36
Degree $2$
Conductor $473$
Sign $0.413 + 0.910i$
Analytic cond. $3.77692$
Root an. cond. $1.94343$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·2-s − 1.84i·3-s + 3.30·4-s − 1.41i·5-s − 4.24i·6-s + 7-s + 3.00·8-s − 0.394·9-s − 3.25i·10-s + (−3 − 1.41i)11-s − 6.08i·12-s + 6.51i·13-s + 2.30·14-s − 2.60·15-s + 0.302·16-s + 0.428i·17-s + ⋯
L(s)  = 1  + 1.62·2-s − 1.06i·3-s + 1.65·4-s − 0.632i·5-s − 1.73i·6-s + 0.377·7-s + 1.06·8-s − 0.131·9-s − 1.02i·10-s + (−0.904 − 0.426i)11-s − 1.75i·12-s + 1.80i·13-s + 0.615·14-s − 0.672·15-s + 0.0756·16-s + 0.103i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $0.413 + 0.910i$
Analytic conductor: \(3.77692\)
Root analytic conductor: \(1.94343\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (472, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 473,\ (\ :1/2),\ 0.413 + 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.83396 - 1.82479i\)
\(L(\frac12)\) \(\approx\) \(2.83396 - 1.82479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (3 + 1.41i)T \)
43 \( 1 + (-5 - 4.24i)T \)
good2 \( 1 - 2.30T + 2T^{2} \)
3 \( 1 + 1.84iT - 3T^{2} \)
5 \( 1 + 1.41iT - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
13 \( 1 - 6.51iT - 13T^{2} \)
17 \( 1 - 0.428iT - 17T^{2} \)
19 \( 1 + 0.394T + 19T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 - 7.60T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 7.49iT - 37T^{2} \)
41 \( 1 + 3.81iT - 41T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 - 1.60T + 53T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 - 4.21T + 61T^{2} \)
67 \( 1 + 2.39T + 67T^{2} \)
71 \( 1 - 9.34iT - 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 1.28iT - 79T^{2} \)
83 \( 1 + 8.05iT - 83T^{2} \)
89 \( 1 + 13.1iT - 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−11.58825314993409844546174214037, −10.22560563823105505426948089585, −8.779629868626009803669525852982, −7.88268717596108236794608249366, −6.74871269037001321900843866825, −6.20940774178026233671834650928, −4.91389680308415909090274990372, −4.34284097890172052390037640476, −2.76490013759328441427785454614, −1.59016805892675114368644140790, 2.63708582034311950503944509241, 3.39360493569368124012090627695, 4.57796859312646286972980107839, 5.14262427933254171367005034606, 6.10430878515351726136082869449, 7.28410004351603991045961480492, 8.298209731794765899443658446063, 9.878225745655592704943948548336, 10.52525388580504790874328219455, 11.08610575302526646689280636620

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