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Properties

Label 2-275-11.3-c1-0-1
Degree 22
Conductor 275275
Sign 0.202−0.979i0.202 - 0.979i
Analytic cond. 2.195882.19588
Root an. cond. 1.481851.48185
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  + (0.0756 − 0.0549i)2-s + (−0.453 − 1.39i)3-s + (−0.615 + 1.89i)4-s + (−0.110 − 0.0806i)6-s + (−1.39 + 4.30i)7-s + (0.115 + 0.354i)8-s + (0.686 − 0.498i)9-s + (−2.39 + 2.29i)11-s + 2.92·12-s + (−0.924 + 0.671i)13-s + (0.130 + 0.402i)14-s + (−3.19 − 2.32i)16-s + (2.72 + 1.98i)17-s + (0.0245 − 0.0754i)18-s + (1.88 + 5.78i)19-s + ⋯
L(s)  = 1  + (0.0534 − 0.0388i)2-s + (−0.261 − 0.805i)3-s + (−0.307 + 0.946i)4-s + (−0.0452 − 0.0329i)6-s + (−0.528 + 1.62i)7-s + (0.0407 + 0.125i)8-s + (0.228 − 0.166i)9-s + (−0.723 + 0.690i)11-s + 0.843·12-s + (−0.256 + 0.186i)13-s + (0.0349 + 0.107i)14-s + (−0.798 − 0.580i)16-s + (0.661 + 0.480i)17-s + (0.00578 − 0.0177i)18-s + (0.431 + 1.32i)19-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=((0.202−0.979i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+1/2) L(s)=((0.202−0.979i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: 0.202−0.979i0.202 - 0.979i
Analytic conductor: 2.195882.19588
Root analytic conductor: 1.481851.48185
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ275(201,⋅)\chi_{275} (201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 275, ( :1/2), 0.202−0.979i)(2,\ 275,\ (\ :1/2),\ 0.202 - 0.979i)

Particular Values

L(1)L(1) ≈\approx 0.716880+0.583937i0.716880 + 0.583937i
L(12)L(\frac12) ≈\approx 0.716880+0.583937i0.716880 + 0.583937i
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)
bad5 1 1
11 1+(2.39−2.29i)T 1 + (2.39 - 2.29i)T
good2 1+(−0.0756+0.0549i)T+(0.618−1.90i)T2 1 + (-0.0756 + 0.0549i)T + (0.618 - 1.90i)T^{2}
3 1+(0.453+1.39i)T+(−2.42+1.76i)T2 1 + (0.453 + 1.39i)T + (-2.42 + 1.76i)T^{2}
7 1+(1.39−4.30i)T+(−5.66−4.11i)T2 1 + (1.39 - 4.30i)T + (-5.66 - 4.11i)T^{2}
13 1+(0.924−0.671i)T+(4.01−12.3i)T2 1 + (0.924 - 0.671i)T + (4.01 - 12.3i)T^{2}
17 1+(−2.72−1.98i)T+(5.25+16.1i)T2 1 + (-2.72 - 1.98i)T + (5.25 + 16.1i)T^{2}
19 1+(−1.88−5.78i)T+(−15.3+11.1i)T2 1 + (-1.88 - 5.78i)T + (-15.3 + 11.1i)T^{2}
23 1−5.45T+23T2 1 - 5.45T + 23T^{2}
29 1+(−1.02+3.15i)T+(−23.4−17.0i)T2 1 + (-1.02 + 3.15i)T + (-23.4 - 17.0i)T^{2}
31 1+(1.44−1.05i)T+(9.57−29.4i)T2 1 + (1.44 - 1.05i)T + (9.57 - 29.4i)T^{2}
37 1+(0.460−1.41i)T+(−29.9−21.7i)T2 1 + (0.460 - 1.41i)T + (-29.9 - 21.7i)T^{2}
41 1+(0.539+1.66i)T+(−33.1+24.0i)T2 1 + (0.539 + 1.66i)T + (-33.1 + 24.0i)T^{2}
43 1−0.263T+43T2 1 - 0.263T + 43T^{2}
47 1+(2.13+6.58i)T+(−38.0+27.6i)T2 1 + (2.13 + 6.58i)T + (-38.0 + 27.6i)T^{2}
53 1+(1.16−0.846i)T+(16.3−50.4i)T2 1 + (1.16 - 0.846i)T + (16.3 - 50.4i)T^{2}
59 1+(2.18−6.72i)T+(−47.7−34.6i)T2 1 + (2.18 - 6.72i)T + (-47.7 - 34.6i)T^{2}
61 1+(2.02+1.47i)T+(18.8+58.0i)T2 1 + (2.02 + 1.47i)T + (18.8 + 58.0i)T^{2}
67 1−0.516T+67T2 1 - 0.516T + 67T^{2}
71 1+(−8.68−6.30i)T+(21.9+67.5i)T2 1 + (-8.68 - 6.30i)T + (21.9 + 67.5i)T^{2}
73 1+(−1.75+5.40i)T+(−59.0−42.9i)T2 1 + (-1.75 + 5.40i)T + (-59.0 - 42.9i)T^{2}
79 1+(−9.14+6.64i)T+(24.4−75.1i)T2 1 + (-9.14 + 6.64i)T + (24.4 - 75.1i)T^{2}
83 1+(−3.62−2.63i)T+(25.6+78.9i)T2 1 + (-3.62 - 2.63i)T + (25.6 + 78.9i)T^{2}
89 1−13.2T+89T2 1 - 13.2T + 89T^{2}
97 1+(2.71−1.97i)T+(29.9−92.2i)T2 1 + (2.71 - 1.97i)T + (29.9 - 92.2i)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

−12.36736989287611428570893347885, −11.69185419288344563001557455022, −10.04564420835930541621027840203, −9.149061242067345826813823612364, −8.118963311586024136903348041332, −7.30450027976084040747263480837, −6.16412048609059406509889259759, −5.08270074147069357330463016892, −3.43810147092618161030688432072, −2.16840713625123104294375783936, 0.73803624662342092181972885555, 3.31695983544725952882665837666, 4.64083677736015246372441103360, 5.27322499330269977424667107120, 6.71029730432573454161725978822, 7.62543017817165213665592676227, 9.266875358281918544838120324592, 9.901343338118355668582599035783, 10.77916486635183925983848003921, 11.04628800916005755215629417291

Graph of the ZZ-function along the critical line