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Properties

Label 1-53-53.50-r1-0-0
Degree $1$
Conductor $53$
Sign $-0.0724 - 0.997i$
Analytic cond. $5.69564$
Root an. cond. $5.69564$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.464 − 0.885i)2-s + (0.935 + 0.354i)3-s + (−0.568 − 0.822i)4-s + (0.663 − 0.748i)5-s + (0.748 − 0.663i)6-s + (−0.885 − 0.464i)7-s + (−0.992 + 0.120i)8-s + (0.748 + 0.663i)9-s + (−0.354 − 0.935i)10-s + (0.970 − 0.239i)11-s + (−0.239 − 0.970i)12-s + (0.568 − 0.822i)13-s + (−0.822 + 0.568i)14-s + (0.885 − 0.464i)15-s + (−0.354 + 0.935i)16-s + (−0.120 + 0.992i)17-s + ⋯
L(s)  = 1  + (0.464 − 0.885i)2-s + (0.935 + 0.354i)3-s + (−0.568 − 0.822i)4-s + (0.663 − 0.748i)5-s + (0.748 − 0.663i)6-s + (−0.885 − 0.464i)7-s + (−0.992 + 0.120i)8-s + (0.748 + 0.663i)9-s + (−0.354 − 0.935i)10-s + (0.970 − 0.239i)11-s + (−0.239 − 0.970i)12-s + (0.568 − 0.822i)13-s + (−0.822 + 0.568i)14-s + (0.885 − 0.464i)15-s + (−0.354 + 0.935i)16-s + (−0.120 + 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0724 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0724 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $-0.0724 - 0.997i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ -0.0724 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.726288783 - 1.856159017i\)
\(L(\frac12)\) \(\approx\) \(1.726288783 - 1.856159017i\)
\(L(1)\) \(\approx\) \(1.493400280 - 0.9489759423i\)
\(L(1)\) \(\approx\) \(1.493400280 - 0.9489759423i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad53 \( 1 \)
good2 \( 1 + (0.464 - 0.885i)T \)
3 \( 1 + (0.935 + 0.354i)T \)
5 \( 1 + (0.663 - 0.748i)T \)
7 \( 1 + (-0.885 - 0.464i)T \)
11 \( 1 + (0.970 - 0.239i)T \)
13 \( 1 + (0.568 - 0.822i)T \)
17 \( 1 + (-0.120 + 0.992i)T \)
19 \( 1 + (-0.822 - 0.568i)T \)
23 \( 1 + iT \)
29 \( 1 + (0.970 + 0.239i)T \)
31 \( 1 + (-0.239 + 0.970i)T \)
37 \( 1 + (0.354 - 0.935i)T \)
41 \( 1 + (0.239 + 0.970i)T \)
43 \( 1 + (0.354 + 0.935i)T \)
47 \( 1 + (-0.748 + 0.663i)T \)
59 \( 1 + (0.748 - 0.663i)T \)
61 \( 1 + (-0.992 + 0.120i)T \)
67 \( 1 + (-0.822 + 0.568i)T \)
71 \( 1 + (0.935 - 0.354i)T \)
73 \( 1 + (-0.992 - 0.120i)T \)
79 \( 1 + (-0.464 - 0.885i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.120 - 0.992i)T \)
97 \( 1 + (-0.748 - 0.663i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−33.165276212767087551486412717340, −32.32951983849496554673589351489, −31.221837200084834631448568958349, −30.32120349292735044122003236776, −29.21538775289216418416493372352, −27.17772799657131764009565820226, −25.98504672998857851179212766578, −25.449297643691450338457672020070, −24.55536186097192426766533175626, −22.99357704633409183759670653951, −22.02085287262140674078992123873, −20.84574434594820593251714107176, −19.05929099883331793397498014669, −18.23789034141238472044601356106, −16.67954736733792963670369317421, −15.28540708746868814798421248040, −14.27843457872534471058697408331, −13.45418738661702868277311613246, −12.13750137941733710301356415448, −9.67059743379113481887876134533, −8.71338214824020373558435316541, −6.9169967249092725091726617739, −6.272525271920534424343677266108, −3.94678409033780381558273231369, −2.526224966309040285529292462406, 1.37631575552000512064565760512, 3.167737202279701417632959283207, 4.37353199011836464192324616797, 6.1495441154390403486361295844, 8.635002483010928965276911935308, 9.58310705839715911657038751150, 10.672249994926160758440049823402, 12.732462665718251209976687945534, 13.373457038744031968700941487662, 14.53746391632666482999721006218, 16.00074623111160853949403028358, 17.60856262700646689100014601763, 19.53436358723526289271678850994, 19.83240414466390454287029620593, 21.19373612219745404347375751596, 21.930773977439727144789975488893, 23.41627403023693800856003937344, 24.84537670859846340992515569811, 25.86074568344269835170021739111, 27.33924670845019970625960477557, 28.30606487866728160465978750574, 29.63945805907900677936059092363, 30.39790587787840357972573220348, 31.84312893736409712608379351373, 32.6490751152542209290505349114

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