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Properties

Label 2-275-11.4-c1-0-1
Degree 22
Conductor 275275
Sign 0.324−0.946i0.324 - 0.946i
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  + (−0.168 − 0.122i)2-s + (−0.585 + 1.80i)3-s + (−0.604 − 1.86i)4-s + (0.319 − 0.231i)6-s + (0.398 + 1.22i)7-s + (−0.254 + 0.783i)8-s + (−0.475 − 0.345i)9-s + (0.898 + 3.19i)11-s + 3.70·12-s + (4.40 + 3.20i)13-s + (0.0829 − 0.255i)14-s + (−3.02 + 2.19i)16-s + (−3.18 + 2.31i)17-s + (0.0378 + 0.116i)18-s + (0.693 − 2.13i)19-s + ⋯
L(s)  = 1  + (−0.119 − 0.0865i)2-s + (−0.337 + 1.04i)3-s + (−0.302 − 0.930i)4-s + (0.130 − 0.0946i)6-s + (0.150 + 0.463i)7-s + (−0.0900 + 0.277i)8-s + (−0.158 − 0.115i)9-s + (0.271 + 0.962i)11-s + 1.06·12-s + (1.22 + 0.888i)13-s + (0.0221 − 0.0682i)14-s + (−0.756 + 0.549i)16-s + (−0.771 + 0.560i)17-s + (0.00892 + 0.0274i)18-s + (0.159 − 0.489i)19-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=((0.324−0.946i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+1/2) L(s)=((0.324−0.946i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: 0.324−0.946i0.324 - 0.946i
Analytic conductor: 2.195882.19588
Root analytic conductor: 1.481851.48185
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ275(26,⋅)\chi_{275} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 275, ( :1/2), 0.324−0.946i)(2,\ 275,\ (\ :1/2),\ 0.324 - 0.946i)

Particular Values

L(1)L(1) ≈\approx 0.819819+0.585730i0.819819 + 0.585730i
L(12)L(\frac12) ≈\approx 0.819819+0.585730i0.819819 + 0.585730i
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+(−0.898−3.19i)T 1 + (-0.898 - 3.19i)T
good2 1+(0.168+0.122i)T+(0.618+1.90i)T2 1 + (0.168 + 0.122i)T + (0.618 + 1.90i)T^{2}
3 1+(0.585−1.80i)T+(−2.42−1.76i)T2 1 + (0.585 - 1.80i)T + (-2.42 - 1.76i)T^{2}
7 1+(−0.398−1.22i)T+(−5.66+4.11i)T2 1 + (-0.398 - 1.22i)T + (-5.66 + 4.11i)T^{2}
13 1+(−4.40−3.20i)T+(4.01+12.3i)T2 1 + (-4.40 - 3.20i)T + (4.01 + 12.3i)T^{2}
17 1+(3.18−2.31i)T+(5.25−16.1i)T2 1 + (3.18 - 2.31i)T + (5.25 - 16.1i)T^{2}
19 1+(−0.693+2.13i)T+(−15.3−11.1i)T2 1 + (-0.693 + 2.13i)T + (-15.3 - 11.1i)T^{2}
23 1−0.711T+23T2 1 - 0.711T + 23T^{2}
29 1+(−1.13−3.47i)T+(−23.4+17.0i)T2 1 + (-1.13 - 3.47i)T + (-23.4 + 17.0i)T^{2}
31 1+(−5.22−3.79i)T+(9.57+29.4i)T2 1 + (-5.22 - 3.79i)T + (9.57 + 29.4i)T^{2}
37 1+(2.55+7.85i)T+(−29.9+21.7i)T2 1 + (2.55 + 7.85i)T + (-29.9 + 21.7i)T^{2}
41 1+(−3.90+12.0i)T+(−33.1−24.0i)T2 1 + (-3.90 + 12.0i)T + (-33.1 - 24.0i)T^{2}
43 1−1.31T+43T2 1 - 1.31T + 43T^{2}
47 1+(1.39−4.29i)T+(−38.0−27.6i)T2 1 + (1.39 - 4.29i)T + (-38.0 - 27.6i)T^{2}
53 1+(7.86+5.71i)T+(16.3+50.4i)T2 1 + (7.86 + 5.71i)T + (16.3 + 50.4i)T^{2}
59 1+(−2.75−8.47i)T+(−47.7+34.6i)T2 1 + (-2.75 - 8.47i)T + (-47.7 + 34.6i)T^{2}
61 1+(−1.48+1.08i)T+(18.8−58.0i)T2 1 + (-1.48 + 1.08i)T + (18.8 - 58.0i)T^{2}
67 1−4.30T+67T2 1 - 4.30T + 67T^{2}
71 1+(8.21−5.97i)T+(21.9−67.5i)T2 1 + (8.21 - 5.97i)T + (21.9 - 67.5i)T^{2}
73 1+(3.70+11.4i)T+(−59.0+42.9i)T2 1 + (3.70 + 11.4i)T + (-59.0 + 42.9i)T^{2}
79 1+(10.3+7.53i)T+(24.4+75.1i)T2 1 + (10.3 + 7.53i)T + (24.4 + 75.1i)T^{2}
83 1+(−10.1+7.37i)T+(25.6−78.9i)T2 1 + (-10.1 + 7.37i)T + (25.6 - 78.9i)T^{2}
89 1+6.28T+89T2 1 + 6.28T + 89T^{2}
97 1+(0.245+0.178i)T+(29.9+92.2i)T2 1 + (0.245 + 0.178i)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

−11.76006403060297775753920342624, −10.85197263746534306301657259121, −10.33709845733294875981977900234, −9.182445435233214540525893574875, −8.836083485224735536456590503449, −6.95103289334238476626213480848, −5.86546961204584027463472957655, −4.81675724868211974765100346771, −4.04953965490504657232711388367, −1.83205248794830684443086280882, 0.929260523826921530712593046165, 3.05429846024216674251380681545, 4.28824219998324538104520404837, 5.99447440560002467932310493882, 6.80170091433284658409990689806, 7.940617135406213439113178853967, 8.399341839466841746435362970605, 9.714417070911193264880816811016, 11.12841431841414873145985958952, 11.71692477270146857840547150696

Graph of the ZZ-function along the critical line