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Properties

Label 2-261-261.101-c1-0-16
Degree 22
Conductor 261261
Sign 0.814−0.580i0.814 - 0.580i
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  + (−1.05 + 2.42i)2-s + (1.16 − 1.27i)3-s + (−3.40 − 3.67i)4-s + (1.73 + 0.261i)5-s + (1.86 + 4.19i)6-s + (−1.57 − 1.45i)7-s + (7.51 − 2.63i)8-s + (−0.262 − 2.98i)9-s + (−2.46 + 3.93i)10-s + (1.73 − 1.49i)11-s + (−8.67 + 0.0558i)12-s + (1.70 − 2.50i)13-s + (5.19 − 2.26i)14-s + (2.36 − 1.90i)15-s + (−0.827 + 11.0i)16-s + (1.44 + 1.44i)17-s + ⋯
L(s)  = 1  + (−0.748 + 1.71i)2-s + (0.675 − 0.737i)3-s + (−1.70 − 1.83i)4-s + (0.775 + 0.116i)5-s + (0.759 + 1.71i)6-s + (−0.593 − 0.550i)7-s + (2.65 − 0.930i)8-s + (−0.0875 − 0.996i)9-s + (−0.780 + 1.24i)10-s + (0.524 − 0.451i)11-s + (−2.50 + 0.0161i)12-s + (0.473 − 0.694i)13-s + (1.38 − 0.606i)14-s + (0.609 − 0.492i)15-s + (−0.206 + 2.76i)16-s + (0.349 + 0.349i)17-s + ⋯

Functional equation

Λ(s)=(261s/2 ΓC(s) L(s)=((0.814−0.580i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(261s/2 ΓC(s+1/2) L(s)=((0.814−0.580i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 261261    =    32⋅293^{2} \cdot 29
Sign: 0.814−0.580i0.814 - 0.580i
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), 0.814−0.580i)(2,\ 261,\ (\ :1/2),\ 0.814 - 0.580i)

Particular Values

L(1)L(1) ≈\approx 1.02212+0.327244i1.02212 + 0.327244i
L(12)L(\frac12) ≈\approx 1.02212+0.327244i1.02212 + 0.327244i
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+(−1.16+1.27i)T 1 + (-1.16 + 1.27i)T
29 1+(3.01+4.46i)T 1 + (3.01 + 4.46i)T
good2 1+(1.05−2.42i)T+(−1.36−1.46i)T2 1 + (1.05 - 2.42i)T + (-1.36 - 1.46i)T^{2}
5 1+(−1.73−0.261i)T+(4.77+1.47i)T2 1 + (-1.73 - 0.261i)T + (4.77 + 1.47i)T^{2}
7 1+(1.57+1.45i)T+(0.523+6.98i)T2 1 + (1.57 + 1.45i)T + (0.523 + 6.98i)T^{2}
11 1+(−1.73+1.49i)T+(1.63−10.8i)T2 1 + (-1.73 + 1.49i)T + (1.63 - 10.8i)T^{2}
13 1+(−1.70+2.50i)T+(−4.74−12.1i)T2 1 + (-1.70 + 2.50i)T + (-4.74 - 12.1i)T^{2}
17 1+(−1.44−1.44i)T+17iT2 1 + (-1.44 - 1.44i)T + 17iT^{2}
19 1+(−6.42−4.03i)T+(8.24+17.1i)T2 1 + (-6.42 - 4.03i)T + (8.24 + 17.1i)T^{2}
23 1+(−1.03−0.407i)T+(16.8+15.6i)T2 1 + (-1.03 - 0.407i)T + (16.8 + 15.6i)T^{2}
31 1+(7.66−5.65i)T+(9.13−29.6i)T2 1 + (7.66 - 5.65i)T + (9.13 - 29.6i)T^{2}
37 1+(1.67+4.79i)T+(−28.9+23.0i)T2 1 + (1.67 + 4.79i)T + (-28.9 + 23.0i)T^{2}
41 1+(−7.61−2.04i)T+(35.5+20.5i)T2 1 + (-7.61 - 2.04i)T + (35.5 + 20.5i)T^{2}
43 1+(1.03−1.40i)T+(−12.6−41.0i)T2 1 + (1.03 - 1.40i)T + (-12.6 - 41.0i)T^{2}
47 1+(−7.94−9.22i)T+(−7.00+46.4i)T2 1 + (-7.94 - 9.22i)T + (-7.00 + 46.4i)T^{2}
53 1+(−3.67−2.93i)T+(11.7+51.6i)T2 1 + (-3.67 - 2.93i)T + (11.7 + 51.6i)T^{2}
59 1+(11.1−6.41i)T+(29.5−51.0i)T2 1 + (11.1 - 6.41i)T + (29.5 - 51.0i)T^{2}
61 1+(12.0−0.450i)T+(60.8−4.55i)T2 1 + (12.0 - 0.450i)T + (60.8 - 4.55i)T^{2}
67 1+(−6.41+0.480i)T+(66.2−9.98i)T2 1 + (-6.41 + 0.480i)T + (66.2 - 9.98i)T^{2}
71 1+(−7.16+3.45i)T+(44.2−55.5i)T2 1 + (-7.16 + 3.45i)T + (44.2 - 55.5i)T^{2}
73 1+(−0.980+8.69i)T+(−71.1−16.2i)T2 1 + (-0.980 + 8.69i)T + (-71.1 - 16.2i)T^{2}
79 1+(−0.814+4.30i)T+(−73.5−28.8i)T2 1 + (-0.814 + 4.30i)T + (-73.5 - 28.8i)T^{2}
83 1+(2.13−6.91i)T+(−68.5−46.7i)T2 1 + (2.13 - 6.91i)T + (-68.5 - 46.7i)T^{2}
89 1+(4.94−0.556i)T+(86.7−19.8i)T2 1 + (4.94 - 0.556i)T + (86.7 - 19.8i)T^{2}
97 1+(−3.14−5.95i)T+(−54.6+80.1i)T2 1 + (-3.14 - 5.95i)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

−12.49047951771865443726798004215, −10.62288771626407724761012812103, −9.566379372076265589713045944290, −9.105427018172791117311471152532, −7.86402106217132953363124554237, −7.35391679717108772647968815673, −6.15418482525324614559383934936, −5.73756936408797513297529610233, −3.61378193080013837695562065967, −1.17813068532803138718297012423, 1.80844517421388211343509165931, 2.94063178418357423828715183167, 3.96038707782298330084330210128, 5.34391044543609681030068485871, 7.38708437318772838424967559048, 8.813498137063979595778473634170, 9.475428117024326463289275612339, 9.601379568417423210336100087278, 10.85047922449469439051814361125, 11.62159239982223845875960962866

Graph of the ZZ-function along the critical line