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Properties

Label 2-15e2-225.106-c1-0-10
Degree 22
Conductor 225225
Sign 0.999+0.0413i0.999 + 0.0413i
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  + (−1.05 − 0.224i)2-s + (1.71 − 0.259i)3-s + (−0.766 − 0.341i)4-s + (−1.13 + 1.92i)5-s + (−1.86 − 0.110i)6-s + (0.316 − 0.548i)7-s + (2.47 + 1.79i)8-s + (2.86 − 0.888i)9-s + (1.63 − 1.77i)10-s + (5.10 + 1.08i)11-s + (−1.40 − 0.385i)12-s + (3.72 − 0.791i)13-s + (−0.456 + 0.506i)14-s + (−1.44 + 3.59i)15-s + (−1.08 − 1.20i)16-s + (−0.365 − 0.265i)17-s + ⋯
L(s)  = 1  + (−0.745 − 0.158i)2-s + (0.988 − 0.149i)3-s + (−0.383 − 0.170i)4-s + (−0.509 + 0.860i)5-s + (−0.760 − 0.0449i)6-s + (0.119 − 0.207i)7-s + (0.874 + 0.635i)8-s + (0.955 − 0.296i)9-s + (0.515 − 0.560i)10-s + (1.53 + 0.327i)11-s + (−0.404 − 0.111i)12-s + (1.03 − 0.219i)13-s + (−0.121 + 0.135i)14-s + (−0.374 + 0.927i)15-s + (−0.270 − 0.300i)16-s + (−0.0886 − 0.0643i)17-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((0.999+0.0413i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0413i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((0.999+0.0413i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0413i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: 0.999+0.0413i0.999 + 0.0413i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ225(106,⋅)\chi_{225} (106, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 225, ( :1/2), 0.999+0.0413i)(2,\ 225,\ (\ :1/2),\ 0.999 + 0.0413i)

Particular Values

L(1)L(1) ≈\approx 1.06294−0.0219952i1.06294 - 0.0219952i
L(12)L(\frac12) ≈\approx 1.06294−0.0219952i1.06294 - 0.0219952i
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+(−1.71+0.259i)T 1 + (-1.71 + 0.259i)T
5 1+(1.13−1.92i)T 1 + (1.13 - 1.92i)T
good2 1+(1.05+0.224i)T+(1.82+0.813i)T2 1 + (1.05 + 0.224i)T + (1.82 + 0.813i)T^{2}
7 1+(−0.316+0.548i)T+(−3.5−6.06i)T2 1 + (-0.316 + 0.548i)T + (-3.5 - 6.06i)T^{2}
11 1+(−5.10−1.08i)T+(10.0+4.47i)T2 1 + (-5.10 - 1.08i)T + (10.0 + 4.47i)T^{2}
13 1+(−3.72+0.791i)T+(11.8−5.28i)T2 1 + (-3.72 + 0.791i)T + (11.8 - 5.28i)T^{2}
17 1+(0.365+0.265i)T+(5.25+16.1i)T2 1 + (0.365 + 0.265i)T + (5.25 + 16.1i)T^{2}
19 1+(3.97+2.88i)T+(5.87+18.0i)T2 1 + (3.97 + 2.88i)T + (5.87 + 18.0i)T^{2}
23 1+(0.0686−0.0762i)T+(−2.40−22.8i)T2 1 + (0.0686 - 0.0762i)T + (-2.40 - 22.8i)T^{2}
29 1+(−1.04−9.98i)T+(−28.3+6.02i)T2 1 + (-1.04 - 9.98i)T + (-28.3 + 6.02i)T^{2}
31 1+(0.422−4.01i)T+(−30.3−6.44i)T2 1 + (0.422 - 4.01i)T + (-30.3 - 6.44i)T^{2}
37 1+(−0.0800+0.246i)T+(−29.9−21.7i)T2 1 + (-0.0800 + 0.246i)T + (-29.9 - 21.7i)T^{2}
41 1+(7.65−1.62i)T+(37.4−16.6i)T2 1 + (7.65 - 1.62i)T + (37.4 - 16.6i)T^{2}
43 1+(−3.68+6.38i)T+(−21.5−37.2i)T2 1 + (-3.68 + 6.38i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.10+10.5i)T+(−45.9+9.77i)T2 1 + (1.10 + 10.5i)T + (-45.9 + 9.77i)T^{2}
53 1+(4.56−3.31i)T+(16.3−50.4i)T2 1 + (4.56 - 3.31i)T + (16.3 - 50.4i)T^{2}
59 1+(7.00−1.48i)T+(53.8−23.9i)T2 1 + (7.00 - 1.48i)T + (53.8 - 23.9i)T^{2}
61 1+(4.50+0.958i)T+(55.7+24.8i)T2 1 + (4.50 + 0.958i)T + (55.7 + 24.8i)T^{2}
67 1+(0.364−3.46i)T+(−65.5−13.9i)T2 1 + (0.364 - 3.46i)T + (-65.5 - 13.9i)T^{2}
71 1+(3.47−2.52i)T+(21.9−67.5i)T2 1 + (3.47 - 2.52i)T + (21.9 - 67.5i)T^{2}
73 1+(4.75+14.6i)T+(−59.0+42.9i)T2 1 + (4.75 + 14.6i)T + (-59.0 + 42.9i)T^{2}
79 1+(0.142+1.35i)T+(−77.2+16.4i)T2 1 + (0.142 + 1.35i)T + (-77.2 + 16.4i)T^{2}
83 1+(−1.95+0.870i)T+(55.5−61.6i)T2 1 + (-1.95 + 0.870i)T + (55.5 - 61.6i)T^{2}
89 1+(5.42+16.7i)T+(−72.0+52.3i)T2 1 + (5.42 + 16.7i)T + (-72.0 + 52.3i)T^{2}
97 1+(−0.637−6.06i)T+(−94.8+20.1i)T2 1 + (-0.637 - 6.06i)T + (-94.8 + 20.1i)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

−12.15441937630736051034893804364, −10.91961526474529891791269219797, −10.28859539592173154442397949998, −8.985773024326805921310094533339, −8.655313692829266957318264885457, −7.39303263726417167351484478073, −6.56959215502592850525230007684, −4.43840697375891569258312741273, −3.41353622791173039220012040174, −1.57177209277507770569990350041, 1.41366533242909221419533272451, 3.81885573122957050813747212555, 4.34876772698173677475504821687, 6.35079011657511941187167871416, 7.82949343407203679442433502010, 8.443632295778827708962925828808, 9.082818217091756030860182598716, 9.807581335527809246580081166054, 11.22703283085274450903246084890, 12.35282211864527302425903378522

Graph of the ZZ-function along the critical line