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Properties

Label 1-53-53.21-r1-0-0
Degree $1$
Conductor $53$
Sign $0.363 + 0.931i$
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  + (−0.822 − 0.568i)2-s + (0.663 + 0.748i)3-s + (0.354 + 0.935i)4-s + (0.992 + 0.120i)5-s + (−0.120 − 0.992i)6-s + (−0.568 + 0.822i)7-s + (0.239 − 0.970i)8-s + (−0.120 + 0.992i)9-s + (−0.748 − 0.663i)10-s + (−0.885 − 0.464i)11-s + (−0.464 + 0.885i)12-s + (−0.354 + 0.935i)13-s + (0.935 − 0.354i)14-s + (0.568 + 0.822i)15-s + (−0.748 + 0.663i)16-s + (0.970 − 0.239i)17-s + ⋯
L(s)  = 1  + (−0.822 − 0.568i)2-s + (0.663 + 0.748i)3-s + (0.354 + 0.935i)4-s + (0.992 + 0.120i)5-s + (−0.120 − 0.992i)6-s + (−0.568 + 0.822i)7-s + (0.239 − 0.970i)8-s + (−0.120 + 0.992i)9-s + (−0.748 − 0.663i)10-s + (−0.885 − 0.464i)11-s + (−0.464 + 0.885i)12-s + (−0.354 + 0.935i)13-s + (0.935 − 0.354i)14-s + (0.568 + 0.822i)15-s + (−0.748 + 0.663i)16-s + (0.970 − 0.239i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.363 + 0.931i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ 0.363 + 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.082198010 + 0.7394831192i\)
\(L(\frac12)\) \(\approx\) \(1.082198010 + 0.7394831192i\)
\(L(1)\) \(\approx\) \(0.9576680180 + 0.2541884710i\)
\(L(1)\) \(\approx\) \(0.9576680180 + 0.2541884710i\)

Euler product

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

−32.80910128692733571925795229390, −32.146688507077753090221796901, −30.28489048143074373681305437764, −29.34753260148508974913617987935, −28.49539433306440315501181826948, −26.71442001887095179118663088713, −25.88169467973248026302705644192, −25.13306812246985615614925469278, −24.04984876803371802184300581784, −22.86745409143687848486317391885, −20.68841371105137054734947279083, −19.954518924786091126607054186475, −18.5505738190513770281152315125, −17.72717800253545028842537958062, −16.56896813716249751866595540728, −14.97354334818028254079554591892, −13.805960650759414338441197164387, −12.736476138845187430396215034730, −10.36747400121802179859078265537, −9.52772643783075476659383619720, −7.97321113119882199820814360341, −6.968575709745799876658426110359, −5.55378303336925183591242876849, −2.649193389629308222487766446706, −0.92264801592774596852173774411, 2.15927970719026244139059752209, 3.30160536662122313040396340967, 5.56898932954641510607008964263, 7.678577346969500808366659430876, 9.31163075774159875969634798544, 9.6700892595449286661752509236, 11.13151666851765269035800764882, 12.82687378084267714062093616394, 14.15339486553868711798181829634, 15.83205573672481818578414747702, 16.73513982416107714004246693990, 18.37034521126047934880313710905, 19.166751516580481164265645915355, 20.6687205045737205597910990928, 21.448685701844394757606168225980, 22.22186672369239418752834179446, 24.699682229378329113174739592365, 25.77603554208259811191598071242, 26.28408870490182264834746400188, 27.63174107615770997295522004479, 28.75294736545697723140247755137, 29.53709560808093034817189446974, 31.15055070219721871159592794955, 31.91320678011426721755146081805, 33.46102555905908886112941191868

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