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Properties

Label 2-15e2-225.31-c1-0-5
Degree 22
Conductor 225225
Sign −0.974−0.224i-0.974 - 0.224i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.727 + 0.807i)2-s + (−0.917 + 1.46i)3-s + (0.0855 + 0.813i)4-s + (0.934 + 2.03i)5-s + (−0.519 − 1.80i)6-s + (1.73 + 3.01i)7-s + (−2.47 − 1.80i)8-s + (−1.31 − 2.69i)9-s + (−2.32 − 0.722i)10-s + (2.35 − 2.61i)11-s + (−1.27 − 0.620i)12-s + (1.45 + 1.61i)13-s + (−3.69 − 0.786i)14-s + (−3.84 − 0.490i)15-s + (1.65 − 0.352i)16-s + (−1.54 − 1.12i)17-s + ⋯
L(s)  = 1  + (−0.514 + 0.571i)2-s + (−0.529 + 0.848i)3-s + (0.0427 + 0.406i)4-s + (0.417 + 0.908i)5-s + (−0.212 − 0.738i)6-s + (0.657 + 1.13i)7-s + (−0.876 − 0.636i)8-s + (−0.439 − 0.898i)9-s + (−0.733 − 0.228i)10-s + (0.709 − 0.788i)11-s + (−0.367 − 0.179i)12-s + (0.402 + 0.447i)13-s + (−0.988 − 0.210i)14-s + (−0.991 − 0.126i)15-s + (0.414 − 0.0880i)16-s + (−0.375 − 0.272i)17-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((−0.974−0.224i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((−0.974−0.224i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: −0.974−0.224i-0.974 - 0.224i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ225(31,⋅)\chi_{225} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 225, ( :1/2), −0.974−0.224i)(2,\ 225,\ (\ :1/2),\ -0.974 - 0.224i)

Particular Values

L(1)L(1) ≈\approx 0.0966030+0.851467i0.0966030 + 0.851467i
L(12)L(\frac12) ≈\approx 0.0966030+0.851467i0.0966030 + 0.851467i
L(32)L(\frac{3}{2}) 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+(0.917−1.46i)T 1 + (0.917 - 1.46i)T
5 1+(−0.934−2.03i)T 1 + (-0.934 - 2.03i)T
good2 1+(0.727−0.807i)T+(−0.209−1.98i)T2 1 + (0.727 - 0.807i)T + (-0.209 - 1.98i)T^{2}
7 1+(−1.73−3.01i)T+(−3.5+6.06i)T2 1 + (-1.73 - 3.01i)T + (-3.5 + 6.06i)T^{2}
11 1+(−2.35+2.61i)T+(−1.14−10.9i)T2 1 + (-2.35 + 2.61i)T + (-1.14 - 10.9i)T^{2}
13 1+(−1.45−1.61i)T+(−1.35+12.9i)T2 1 + (-1.45 - 1.61i)T + (-1.35 + 12.9i)T^{2}
17 1+(1.54+1.12i)T+(5.25+16.1i)T2 1 + (1.54 + 1.12i)T + (5.25 + 16.1i)T^{2}
19 1+(0.970+0.704i)T+(5.87+18.0i)T2 1 + (0.970 + 0.704i)T + (5.87 + 18.0i)T^{2}
23 1+(1.96+0.417i)T+(21.0+9.35i)T2 1 + (1.96 + 0.417i)T + (21.0 + 9.35i)T^{2}
29 1+(3.26+1.45i)T+(19.4+21.5i)T2 1 + (3.26 + 1.45i)T + (19.4 + 21.5i)T^{2}
31 1+(−5.79+2.58i)T+(20.7−23.0i)T2 1 + (-5.79 + 2.58i)T + (20.7 - 23.0i)T^{2}
37 1+(−2.46+7.58i)T+(−29.9−21.7i)T2 1 + (-2.46 + 7.58i)T + (-29.9 - 21.7i)T^{2}
41 1+(−1.73−1.92i)T+(−4.28+40.7i)T2 1 + (-1.73 - 1.92i)T + (-4.28 + 40.7i)T^{2}
43 1+(−5.34−9.26i)T+(−21.5+37.2i)T2 1 + (-5.34 - 9.26i)T + (-21.5 + 37.2i)T^{2}
47 1+(−11.9−5.34i)T+(31.4+34.9i)T2 1 + (-11.9 - 5.34i)T + (31.4 + 34.9i)T^{2}
53 1+(11.2−8.18i)T+(16.3−50.4i)T2 1 + (11.2 - 8.18i)T + (16.3 - 50.4i)T^{2}
59 1+(3.64+4.04i)T+(−6.16+58.6i)T2 1 + (3.64 + 4.04i)T + (-6.16 + 58.6i)T^{2}
61 1+(−0.353+0.392i)T+(−6.37−60.6i)T2 1 + (-0.353 + 0.392i)T + (-6.37 - 60.6i)T^{2}
67 1+(9.83−4.37i)T+(44.8−49.7i)T2 1 + (9.83 - 4.37i)T + (44.8 - 49.7i)T^{2}
71 1+(−6.38+4.63i)T+(21.9−67.5i)T2 1 + (-6.38 + 4.63i)T + (21.9 - 67.5i)T^{2}
73 1+(2.75+8.46i)T+(−59.0+42.9i)T2 1 + (2.75 + 8.46i)T + (-59.0 + 42.9i)T^{2}
79 1+(−6.17−2.74i)T+(52.8+58.7i)T2 1 + (-6.17 - 2.74i)T + (52.8 + 58.7i)T^{2}
83 1+(−0.372+3.53i)T+(−81.1−17.2i)T2 1 + (-0.372 + 3.53i)T + (-81.1 - 17.2i)T^{2}
89 1+(−2.71−8.36i)T+(−72.0+52.3i)T2 1 + (-2.71 - 8.36i)T + (-72.0 + 52.3i)T^{2}
97 1+(15.6+6.98i)T+(64.9+72.0i)T2 1 + (15.6 + 6.98i)T + (64.9 + 72.0i)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

−12.36238202482677845519397460675, −11.46902159999408117311117898626, −10.93684691554729500810331688312, −9.392272836234706401852527372940, −9.029548192692308482652555016022, −7.80577669542527337066025113444, −6.31866046348892520088068436004, −5.93015416532149255749269629357, −4.17823627169106997356117649982, −2.78559570253842474530398484169, 0.971763339454350420492917129443, 1.90477935073733786095969733901, 4.43695632030613276534851291474, 5.59264648539826489941502259288, 6.68382813077524476658892387748, 7.932468704386456032129577694301, 8.890703104325643878203167592887, 10.09971252149568245644673899674, 10.78596986566261388129312930720, 11.76865463406391166206515741429

Graph of the ZZ-function along the critical line