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Properties

Label 1-53-53.5-r1-0-0
Degree $1$
Conductor $53$
Sign $-0.952 - 0.305i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $-0.952 - 0.305i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ -0.952 - 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1047928025 - 0.6699669126i\)
\(L(\frac12)\) \(\approx\) \(0.1047928025 - 0.6699669126i\)
\(L(1)\) \(\approx\) \(0.7754308242 - 0.3457137959i\)
\(L(1)\) \(\approx\) \(0.7754308242 - 0.3457137959i\)

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

−34.03583220008887881973438025597, −32.1314401556898646673051195696, −31.46960114457424416230190121821, −30.423610594486825909136080072315, −29.20438110644794172603711187671, −28.13389074172378551315728019961, −26.52256124848081960752174790979, −25.23141321389782388477178347830, −24.08487917419890211331206843740, −23.42404269918631018647176655197, −22.40015097182236545414045366089, −21.227072775681946971012887331, −19.38490650842811654422617692350, −18.41439721177326820465925141235, −16.64876076490787199489584380498, −16.00115332869806762943733747248, −14.57035495490787397572447387917, −12.937862856673443144177258276921, −12.23867677420654475800037569843, −11.139548082767593066386893326381, −8.4862893640823804937771257072, −7.28251589171174861529513463077, −6.05076418883652847793803433135, −4.71450495567369449752446708656, −2.76923130539153713845782555232, 0.305285977139284545750098751734, 3.29783991343050627754216903439, 4.33697766637913659629739266049, 5.74168107014996955327492949805, 7.44930826591913814260634138814, 9.97434939085937462678855952241, 10.69051750885748576977780918615, 11.99554897672051763610049976336, 13.05671921281000614511597643999, 14.92536678828704703393522739634, 15.6399171590791532309698074301, 16.95399959488983212825197004021, 18.82059160278862423126891545684, 20.08824985839481762654030729513, 20.95128299520458474441272668183, 22.40375031561439688620273756114, 23.16551903416427532414403982874, 23.7751219405938499590816277047, 25.84798398885474011454018814470, 27.28612982678636633350847834956, 28.03088590921762686612040703730, 29.31145248587206470973695118385, 30.19653846933446100497386145220, 31.73520343293829243012216100228, 32.28582729675497557294841519447

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