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Properties

Label 2-15e2-225.31-c1-0-13
Degree 22
Conductor 225225
Sign 0.940+0.338i0.940 + 0.338i
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  + (0.797 − 0.885i)2-s + (−1.56 + 0.733i)3-s + (0.0604 + 0.575i)4-s + (1.30 − 1.81i)5-s + (−0.602 + 1.97i)6-s + (0.157 + 0.272i)7-s + (2.48 + 1.80i)8-s + (1.92 − 2.30i)9-s + (−0.566 − 2.60i)10-s + (1.93 − 2.14i)11-s + (−0.516 − 0.858i)12-s + (4.36 + 4.85i)13-s + (0.367 + 0.0781i)14-s + (−0.719 + 3.80i)15-s + (2.45 − 0.521i)16-s + (0.794 + 0.577i)17-s + ⋯
L(s)  = 1  + (0.564 − 0.626i)2-s + (−0.906 + 0.423i)3-s + (0.0302 + 0.287i)4-s + (0.584 − 0.811i)5-s + (−0.245 + 0.806i)6-s + (0.0595 + 0.103i)7-s + (0.879 + 0.638i)8-s + (0.641 − 0.766i)9-s + (−0.179 − 0.823i)10-s + (0.583 − 0.647i)11-s + (−0.149 − 0.247i)12-s + (1.21 + 1.34i)13-s + (0.0982 + 0.0208i)14-s + (−0.185 + 0.982i)15-s + (0.613 − 0.130i)16-s + (0.192 + 0.139i)17-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((0.940+0.338i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((0.940+0.338i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: 0.940+0.338i0.940 + 0.338i
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), 0.940+0.338i)(2,\ 225,\ (\ :1/2),\ 0.940 + 0.338i)

Particular Values

L(1)L(1) ≈\approx 1.45054−0.253220i1.45054 - 0.253220i
L(12)L(\frac12) ≈\approx 1.45054−0.253220i1.45054 - 0.253220i
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+(1.56−0.733i)T 1 + (1.56 - 0.733i)T
5 1+(−1.30+1.81i)T 1 + (-1.30 + 1.81i)T
good2 1+(−0.797+0.885i)T+(−0.209−1.98i)T2 1 + (-0.797 + 0.885i)T + (-0.209 - 1.98i)T^{2}
7 1+(−0.157−0.272i)T+(−3.5+6.06i)T2 1 + (-0.157 - 0.272i)T + (-3.5 + 6.06i)T^{2}
11 1+(−1.93+2.14i)T+(−1.14−10.9i)T2 1 + (-1.93 + 2.14i)T + (-1.14 - 10.9i)T^{2}
13 1+(−4.36−4.85i)T+(−1.35+12.9i)T2 1 + (-4.36 - 4.85i)T + (-1.35 + 12.9i)T^{2}
17 1+(−0.794−0.577i)T+(5.25+16.1i)T2 1 + (-0.794 - 0.577i)T + (5.25 + 16.1i)T^{2}
19 1+(5.88+4.27i)T+(5.87+18.0i)T2 1 + (5.88 + 4.27i)T + (5.87 + 18.0i)T^{2}
23 1+(−1.28−0.272i)T+(21.0+9.35i)T2 1 + (-1.28 - 0.272i)T + (21.0 + 9.35i)T^{2}
29 1+(3.94+1.75i)T+(19.4+21.5i)T2 1 + (3.94 + 1.75i)T + (19.4 + 21.5i)T^{2}
31 1+(7.91−3.52i)T+(20.7−23.0i)T2 1 + (7.91 - 3.52i)T + (20.7 - 23.0i)T^{2}
37 1+(−0.00738+0.0227i)T+(−29.9−21.7i)T2 1 + (-0.00738 + 0.0227i)T + (-29.9 - 21.7i)T^{2}
41 1+(−3.06−3.40i)T+(−4.28+40.7i)T2 1 + (-3.06 - 3.40i)T + (-4.28 + 40.7i)T^{2}
43 1+(1.34+2.33i)T+(−21.5+37.2i)T2 1 + (1.34 + 2.33i)T + (-21.5 + 37.2i)T^{2}
47 1+(4.76+2.12i)T+(31.4+34.9i)T2 1 + (4.76 + 2.12i)T + (31.4 + 34.9i)T^{2}
53 1+(3.44−2.50i)T+(16.3−50.4i)T2 1 + (3.44 - 2.50i)T + (16.3 - 50.4i)T^{2}
59 1+(−3.46−3.84i)T+(−6.16+58.6i)T2 1 + (-3.46 - 3.84i)T + (-6.16 + 58.6i)T^{2}
61 1+(4.95−5.50i)T+(−6.37−60.6i)T2 1 + (4.95 - 5.50i)T + (-6.37 - 60.6i)T^{2}
67 1+(2.27−1.01i)T+(44.8−49.7i)T2 1 + (2.27 - 1.01i)T + (44.8 - 49.7i)T^{2}
71 1+(−9.63+7.00i)T+(21.9−67.5i)T2 1 + (-9.63 + 7.00i)T + (21.9 - 67.5i)T^{2}
73 1+(0.283+0.872i)T+(−59.0+42.9i)T2 1 + (0.283 + 0.872i)T + (-59.0 + 42.9i)T^{2}
79 1+(10.3+4.62i)T+(52.8+58.7i)T2 1 + (10.3 + 4.62i)T + (52.8 + 58.7i)T^{2}
83 1+(−1.03+9.88i)T+(−81.1−17.2i)T2 1 + (-1.03 + 9.88i)T + (-81.1 - 17.2i)T^{2}
89 1+(−3.78−11.6i)T+(−72.0+52.3i)T2 1 + (-3.78 - 11.6i)T + (-72.0 + 52.3i)T^{2}
97 1+(6.75+3.00i)T+(64.9+72.0i)T2 1 + (6.75 + 3.00i)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

−12.10662568912641666239290045990, −11.30746778569059809339962997235, −10.73669845571165229512577886658, −9.239921532746259974039793312548, −8.616595112729179873715562725994, −6.78500717750421154326544100337, −5.76640892117427536103499700401, −4.58484336433911971142553325720, −3.77790065659370990435504570904, −1.67777899600964975604989546505, 1.65064791605194197304253738365, 3.93607526901711181379964232576, 5.44630874779649818838713589039, 6.07046247639122664383866718575, 6.86075133844797830731214218668, 7.85462267391715482288484715704, 9.679735711954971087205908912211, 10.66475165602129073135317847377, 11.04747751980182670943521411836, 12.68805904856569402679844836819

Graph of the ZZ-function along the critical line