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Properties

Label 1-53-53.3-r1-0-0
Degree $1$
Conductor $53$
Sign $0.974 + 0.223i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.974 + 0.223i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ 0.974 + 0.223i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6520215181 + 0.07368321742i\)
\(L(\frac12)\) \(\approx\) \(0.6520215181 + 0.07368321742i\)
\(L(1)\) \(\approx\) \(0.5616792506 + 0.1337864386i\)
\(L(1)\) \(\approx\) \(0.5616792506 + 0.1337864386i\)

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

−32.86758732305956094774794649439, −31.82914700262738242036535887014, −30.64848305321330307529042223381, −29.16245865348261517181041213228, −28.53795497417586848549755560256, −27.64408509698383129879425741715, −26.71857324314472984500184666599, −25.13809874315550571974268085954, −23.45434361877314586081042800659, −22.466112294975764040193574027125, −21.47326440306048724280080607217, −20.15812724219843410397963382648, −19.11377607204480975720562022097, −17.81121938741046804410668036062, −16.54477872593990623225208791310, −15.81576096753828542963953671610, −13.37223934259121379014288757267, −11.93217260050464151694703400533, −11.61773906104938398744835432012, −9.75565855673807396025602286463, −8.93322735665775311777423070736, −6.8863799965698421056089660932, −4.87665022653219787225984091157, −3.57310421837356414602630288779, −0.99556171507493885097412665946, 0.70348090182065996456245498293, 3.97104035173723032165264670571, 6.01472429153184783679318541306, 6.781804532812151577860782509608, 8.05985051822510659142886274130, 10.029648352030280323598427784092, 11.04139414699101542300972152580, 12.71360161385347455822517850905, 14.21988611405289132978131104985, 15.68716350996335423344670479012, 16.56759759432066821041029934414, 17.74189024223082961955061900653, 18.85609878790610369991124434629, 19.719799261559173669966714358536, 22.39247553119063226257846503581, 22.773662708426880937653882247186, 23.86471599900087238612202211961, 25.068562670967316639857521143114, 26.366577831452915428852861257521, 27.3346466799053395884772586749, 28.34414456155551977549656150713, 29.61899536161661579903965319127, 30.732690730442239783999524942028, 32.51733017242163942417733347521, 33.21033919235834912210207186122

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