↑

Properties

Label 1-73-73.72-r0-0-0
Degree 11
Conductor 7373
Sign 11
Analytic cond. 0.3390100.339010
Root an. cond. 0.3390100.339010
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s − 11-s + 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 18-s + 19-s − 20-s − 21-s − 22-s + 23-s + 24-s + 25-s − 26-s + 27-s − 28-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s − 11-s + 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 18-s + 19-s − 20-s − 21-s − 22-s + 23-s + 24-s + 25-s − 26-s + 27-s − 28-s + ⋯

Functional equation

Λ(s)=(73s/2 ΓR(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(73s/2 ΓR(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 7373
Sign: 11
Analytic conductor: 0.3390100.339010
Root analytic conductor: 0.3390100.339010
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ73(72,⋅)\chi_{73} (72, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (1, 73, (0: ), 1)(1,\ 73,\ (0:\ ),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7419074301.741907430
L(12)L(\frac12) ≈\approx 1.7419074301.741907430
L(1)L(1) ≈\approx 1.7946364831.794636483
L(1)L(1) ≈\approx 1.7946364831.794636483

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad73 1 1
good2 1+T 1 + T
3 1+T 1 + T
5 1−T 1 - T
7 1−T 1 - T
11 1−T 1 - T
13 1−T 1 - T
17 1−T 1 - T
19 1+T 1 + T
23 1+T 1 + T
29 1−T 1 - T
31 1−T 1 - T
37 1+T 1 + T
41 1+T 1 + T
43 1−T 1 - T
47 1−T 1 - T
53 1−T 1 - T
59 1−T 1 - T
61 1+T 1 + T
67 1+T 1 + T
71 1+T 1 + T
79 1+T 1 + T
83 1−T 1 - T
89 1+T 1 + T
97 1+T 1 + T
show more
show less
   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−31.410952780342472027363132148771, −31.02736170697278072454381676429, −29.631583079970470978967117267366, −28.659981232933365653871089618185, −26.84786900747086139196535056632, −26.09180709583076451803961596351, −24.81571716970897524236893595400, −23.95287701989959187369445244593, −22.790404010198814653325005383796, −21.76766221918046631972443659677, −20.3432287181759549011681855680, −19.772275960075922407417433471963, −18.722161242359284092833550208002, −16.36169750261631524140625675010, −15.53955622545684648307412950081, −14.71339103427956601756490939006, −13.23662076809175706007294086306, −12.61039723044271894373865255190, −11.057600111831301161790763704769, −9.538462577126519392220210694474, −7.78599828751643495749040391811, −6.94757901318678836889055748081, −4.918936476482390378102427375243, −3.558510452056881428407611436422, −2.59808063262299142208016507350, 2.59808063262299142208016507350, 3.558510452056881428407611436422, 4.918936476482390378102427375243, 6.94757901318678836889055748081, 7.78599828751643495749040391811, 9.538462577126519392220210694474, 11.057600111831301161790763704769, 12.61039723044271894373865255190, 13.23662076809175706007294086306, 14.71339103427956601756490939006, 15.53955622545684648307412950081, 16.36169750261631524140625675010, 18.722161242359284092833550208002, 19.772275960075922407417433471963, 20.3432287181759549011681855680, 21.76766221918046631972443659677, 22.790404010198814653325005383796, 23.95287701989959187369445244593, 24.81571716970897524236893595400, 26.09180709583076451803961596351, 26.84786900747086139196535056632, 28.659981232933365653871089618185, 29.631583079970470978967117267366, 31.02736170697278072454381676429, 31.410952780342472027363132148771

Graph of the ZZ-function along the critical line