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Properties

Label 2-473-473.472-c1-0-38
Degree $2$
Conductor $473$
Sign $-0.483 - 0.875i$
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  − 1.16·2-s − 3.00i·3-s − 0.631·4-s − 3.57i·5-s + 3.51i·6-s − 4.51·7-s + 3.07·8-s − 6.01·9-s + 4.18i·10-s + (2.59 − 2.06i)11-s + 1.89i·12-s − 1.89i·13-s + 5.28·14-s − 10.7·15-s − 2.33·16-s − 2.91i·17-s + ⋯
L(s)  = 1  − 0.827·2-s − 1.73i·3-s − 0.315·4-s − 1.60i·5-s + 1.43i·6-s − 1.70·7-s + 1.08·8-s − 2.00·9-s + 1.32i·10-s + (0.781 − 0.623i)11-s + 0.547i·12-s − 0.524i·13-s + 1.41·14-s − 2.77·15-s − 0.584·16-s − 0.706i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $-0.483 - 0.875i$
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: \((2,\ 473,\ (\ :1/2),\ -0.483 - 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.264501 + 0.448501i\)
\(L(\frac12)\) \(\approx\) \(0.264501 + 0.448501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-2.59 + 2.06i)T \)
43 \( 1 + (1.09 - 6.46i)T \)
good2 \( 1 + 1.16T + 2T^{2} \)
3 \( 1 + 3.00iT - 3T^{2} \)
5 \( 1 + 3.57iT - 5T^{2} \)
7 \( 1 + 4.51T + 7T^{2} \)
13 \( 1 + 1.89iT - 13T^{2} \)
17 \( 1 + 2.91iT - 17T^{2} \)
19 \( 1 + 0.0511T + 19T^{2} \)
23 \( 1 - 4.05T + 23T^{2} \)
29 \( 1 - 8.95T + 29T^{2} \)
31 \( 1 - 8.88T + 31T^{2} \)
37 \( 1 + 2.19iT - 37T^{2} \)
41 \( 1 + 1.53iT - 41T^{2} \)
47 \( 1 + 7.79T + 47T^{2} \)
53 \( 1 + 7.76T + 53T^{2} \)
59 \( 1 - 4.38T + 59T^{2} \)
61 \( 1 + 7.38T + 61T^{2} \)
67 \( 1 + 0.464T + 67T^{2} \)
71 \( 1 - 9.21iT - 71T^{2} \)
73 \( 1 - 2.44T + 73T^{2} \)
79 \( 1 + 5.76iT - 79T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 + 8.70iT - 89T^{2} \)
97 \( 1 - 6.02T + 97T^{2} \)
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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

−10.04739707510145383974497632267, −9.199322505212010125268733300456, −8.622979626346448659196065096811, −7.918801708676629498964202073583, −6.80548168744673024845919475877, −6.06120503522108957183166213466, −4.76277653213184106499274248762, −3.03307541134846464120698215471, −1.15239923186541023559523650531, −0.51576796029467416864065926512, 2.90763606781809775753620280503, 3.71879530123413330846905598752, 4.65240388572046685912967757859, 6.32757560035703546054063424437, 6.85671931495879799720544375620, 8.399694605174205775014760012223, 9.401139151908575297494322241300, 9.837488445721870585426757954584, 10.33704448738077674084465284007, 11.00352133496664740532365911613

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