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Properties

Label 2-2523-87.5-c0-0-3
Degree 22
Conductor 25232523
Sign 0.944+0.328i0.944 + 0.328i
Analytic cond. 1.259141.25914
Root an. cond. 1.122111.12211
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.974 − 0.222i)3-s + (0.222 − 0.974i)4-s + (0.360 + 1.57i)7-s + (0.900 − 0.433i)9-s − i·12-s + (0.556 + 0.268i)13-s + (−0.900 − 0.433i)16-s + (−0.602 − 0.137i)19-s + (0.702 + 1.45i)21-s + (−0.222 + 0.974i)25-s + (0.781 − 0.623i)27-s + 1.61·28-s + (1.26 − 1.00i)31-s + (−0.222 − 0.974i)36-s + (0.268 + 0.556i)37-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)3-s + (0.222 − 0.974i)4-s + (0.360 + 1.57i)7-s + (0.900 − 0.433i)9-s − i·12-s + (0.556 + 0.268i)13-s + (−0.900 − 0.433i)16-s + (−0.602 − 0.137i)19-s + (0.702 + 1.45i)21-s + (−0.222 + 0.974i)25-s + (0.781 − 0.623i)27-s + 1.61·28-s + (1.26 − 1.00i)31-s + (−0.222 − 0.974i)36-s + (0.268 + 0.556i)37-s + ⋯

Functional equation

Λ(s)=(2523s/2 ΓC(s) L(s)=((0.944+0.328i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2523s/2 ΓC(s) L(s)=((0.944+0.328i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25232523    =    3⋅2923 \cdot 29^{2}
Sign: 0.944+0.328i0.944 + 0.328i
Analytic conductor: 1.259141.25914
Root analytic conductor: 1.122111.12211
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2523(1745,⋅)\chi_{2523} (1745, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2523, ( :0), 0.944+0.328i)(2,\ 2523,\ (\ :0),\ 0.944 + 0.328i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8961540591.896154059
L(12)L(\frac12) ≈\approx 1.8961540591.896154059
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−0.974+0.222i)T 1 + (-0.974 + 0.222i)T
29 1 1
good2 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
5 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
7 1+(−0.360−1.57i)T+(−0.900+0.433i)T2 1 + (-0.360 - 1.57i)T + (-0.900 + 0.433i)T^{2}
11 1+(0.623+0.781i)T2 1 + (0.623 + 0.781i)T^{2}
13 1+(−0.556−0.268i)T+(0.623+0.781i)T2 1 + (-0.556 - 0.268i)T + (0.623 + 0.781i)T^{2}
17 1+T2 1 + T^{2}
19 1+(0.602+0.137i)T+(0.900+0.433i)T2 1 + (0.602 + 0.137i)T + (0.900 + 0.433i)T^{2}
23 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
31 1+(−1.26+1.00i)T+(0.222−0.974i)T2 1 + (-1.26 + 1.00i)T + (0.222 - 0.974i)T^{2}
37 1+(−0.268−0.556i)T+(−0.623+0.781i)T2 1 + (-0.268 - 0.556i)T + (-0.623 + 0.781i)T^{2}
41 1+T2 1 + T^{2}
43 1+(1.26+1.00i)T+(0.222+0.974i)T2 1 + (1.26 + 1.00i)T + (0.222 + 0.974i)T^{2}
47 1+(0.623+0.781i)T2 1 + (0.623 + 0.781i)T^{2}
53 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
59 1−T2 1 - T^{2}
61 1+(1.57−0.360i)T+(0.900−0.433i)T2 1 + (1.57 - 0.360i)T + (0.900 - 0.433i)T^{2}
67 1+(−0.556+0.268i)T+(0.623−0.781i)T2 1 + (-0.556 + 0.268i)T + (0.623 - 0.781i)T^{2}
71 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
73 1+(0.483+0.385i)T+(0.222+0.974i)T2 1 + (0.483 + 0.385i)T + (0.222 + 0.974i)T^{2}
79 1+(0.268+0.556i)T+(−0.623+0.781i)T2 1 + (0.268 + 0.556i)T + (-0.623 + 0.781i)T^{2}
83 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
89 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
97 1+(1.57+0.360i)T+(0.900+0.433i)T2 1 + (1.57 + 0.360i)T + (0.900 + 0.433i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.967809235738652347650844027306, −8.509888437963541001381321158505, −7.66008780337340291038025996551, −6.60559721929723431610654057069, −6.06015284112235875742437785966, −5.19852358257470810404085787702, −4.32350570855367022798887020052, −3.05797101259380730783350651763, −2.21774117122829491258543916756, −1.51110283172942226236013210585, 1.42468404928078472860675137237, 2.66441359374599962459064455548, 3.50850938555838022659232622511, 4.18373609576692480589597194710, 4.74494793862872465053402045432, 6.41969587254055375828543162784, 7.00329417472932272377337253556, 7.901070832844520024600377113920, 8.139610309910418440356872090193, 8.905303543064560405346623516744

Graph of the ZZ-function along the critical line