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Properties

Label 2-261-261.101-c1-0-2
Degree 22
Conductor 261261
Sign −0.929−0.368i-0.929 - 0.368i
Analytic cond. 2.084092.08409
Root an. cond. 1.443631.44363
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  + (−0.234 + 0.536i)2-s + (−1.27 + 1.17i)3-s + (1.12 + 1.21i)4-s + (−1.79 − 0.269i)5-s + (−0.334 − 0.957i)6-s + (2.59 + 2.40i)7-s + (−2.02 + 0.707i)8-s + (0.225 − 2.99i)9-s + (0.564 − 0.898i)10-s + (−0.974 + 0.838i)11-s + (−2.86 − 0.215i)12-s + (−2.06 + 3.02i)13-s + (−1.89 + 0.828i)14-s + (2.59 − 1.76i)15-s + (−0.154 + 2.05i)16-s + (−2.82 − 2.82i)17-s + ⋯
L(s)  = 1  + (−0.165 + 0.379i)2-s + (−0.733 + 0.679i)3-s + (0.563 + 0.607i)4-s + (−0.801 − 0.120i)5-s + (−0.136 − 0.390i)6-s + (0.980 + 0.909i)7-s + (−0.714 + 0.250i)8-s + (0.0752 − 0.997i)9-s + (0.178 − 0.283i)10-s + (−0.293 + 0.252i)11-s + (−0.826 − 0.0621i)12-s + (−0.571 + 0.838i)13-s + (−0.507 + 0.221i)14-s + (0.669 − 0.456i)15-s + (−0.0385 + 0.513i)16-s + (−0.684 − 0.684i)17-s + ⋯

Functional equation

Λ(s)=(261s/2 ΓC(s) L(s)=((−0.929−0.368i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(261s/2 ΓC(s+1/2) L(s)=((−0.929−0.368i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 261261    =    32⋅293^{2} \cdot 29
Sign: −0.929−0.368i-0.929 - 0.368i
Analytic conductor: 2.084092.08409
Root analytic conductor: 1.443631.44363
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ261(101,⋅)\chi_{261} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 261, ( :1/2), −0.929−0.368i)(2,\ 261,\ (\ :1/2),\ -0.929 - 0.368i)

Particular Values

L(1)L(1) ≈\approx 0.147081+0.771116i0.147081 + 0.771116i
L(12)L(\frac12) ≈\approx 0.147081+0.771116i0.147081 + 0.771116i
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+(1.27−1.17i)T 1 + (1.27 - 1.17i)T
29 1+(1.28−5.23i)T 1 + (1.28 - 5.23i)T
good2 1+(0.234−0.536i)T+(−1.36−1.46i)T2 1 + (0.234 - 0.536i)T + (-1.36 - 1.46i)T^{2}
5 1+(1.79+0.269i)T+(4.77+1.47i)T2 1 + (1.79 + 0.269i)T + (4.77 + 1.47i)T^{2}
7 1+(−2.59−2.40i)T+(0.523+6.98i)T2 1 + (-2.59 - 2.40i)T + (0.523 + 6.98i)T^{2}
11 1+(0.974−0.838i)T+(1.63−10.8i)T2 1 + (0.974 - 0.838i)T + (1.63 - 10.8i)T^{2}
13 1+(2.06−3.02i)T+(−4.74−12.1i)T2 1 + (2.06 - 3.02i)T + (-4.74 - 12.1i)T^{2}
17 1+(2.82+2.82i)T+17iT2 1 + (2.82 + 2.82i)T + 17iT^{2}
19 1+(5.81+3.65i)T+(8.24+17.1i)T2 1 + (5.81 + 3.65i)T + (8.24 + 17.1i)T^{2}
23 1+(−8.33−3.27i)T+(16.8+15.6i)T2 1 + (-8.33 - 3.27i)T + (16.8 + 15.6i)T^{2}
31 1+(−3.72+2.74i)T+(9.13−29.6i)T2 1 + (-3.72 + 2.74i)T + (9.13 - 29.6i)T^{2}
37 1+(−2.62−7.49i)T+(−28.9+23.0i)T2 1 + (-2.62 - 7.49i)T + (-28.9 + 23.0i)T^{2}
41 1+(−6.95−1.86i)T+(35.5+20.5i)T2 1 + (-6.95 - 1.86i)T + (35.5 + 20.5i)T^{2}
43 1+(−1.32+1.79i)T+(−12.6−41.0i)T2 1 + (-1.32 + 1.79i)T + (-12.6 - 41.0i)T^{2}
47 1+(−1.08−1.26i)T+(−7.00+46.4i)T2 1 + (-1.08 - 1.26i)T + (-7.00 + 46.4i)T^{2}
53 1+(−6.55−5.22i)T+(11.7+51.6i)T2 1 + (-6.55 - 5.22i)T + (11.7 + 51.6i)T^{2}
59 1+(7.47−4.31i)T+(29.5−51.0i)T2 1 + (7.47 - 4.31i)T + (29.5 - 51.0i)T^{2}
61 1+(−13.5+0.506i)T+(60.8−4.55i)T2 1 + (-13.5 + 0.506i)T + (60.8 - 4.55i)T^{2}
67 1+(−10.9+0.819i)T+(66.2−9.98i)T2 1 + (-10.9 + 0.819i)T + (66.2 - 9.98i)T^{2}
71 1+(3.89−1.87i)T+(44.2−55.5i)T2 1 + (3.89 - 1.87i)T + (44.2 - 55.5i)T^{2}
73 1+(−0.963+8.55i)T+(−71.1−16.2i)T2 1 + (-0.963 + 8.55i)T + (-71.1 - 16.2i)T^{2}
79 1+(−0.335+1.77i)T+(−73.5−28.8i)T2 1 + (-0.335 + 1.77i)T + (-73.5 - 28.8i)T^{2}
83 1+(0.302−0.980i)T+(−68.5−46.7i)T2 1 + (0.302 - 0.980i)T + (-68.5 - 46.7i)T^{2}
89 1+(2.15−0.242i)T+(86.7−19.8i)T2 1 + (2.15 - 0.242i)T + (86.7 - 19.8i)T^{2}
97 1+(0.817+1.54i)T+(−54.6+80.1i)T2 1 + (0.817 + 1.54i)T + (-54.6 + 80.1i)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

−11.99765212591360249588545360064, −11.46946064434968153256332837814, −10.94072792563807607621451041919, −9.251632813132901303823704813069, −8.609721260005858404931029280654, −7.41872313239674267594784390235, −6.54260482425345292420371964112, −5.12863478772002369810442191930, −4.31127203550797160037291547683, −2.58891434033142035697080879119, 0.69096279217020225538143137256, 2.27307466641504418426603724157, 4.25040619474159515914671422724, 5.49730019719160103496423602931, 6.65512302626549142370160792103, 7.56351146080921854858517302192, 8.357519393612497433064573199023, 10.20559349525106568395890140543, 10.97127094190882966852468836910, 11.18441922195657589821509005759

Graph of the ZZ-function along the critical line