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Properties

Label 2-130-13.10-c1-0-1
Degree 22
Conductor 130130
Sign 0.964−0.265i0.964 - 0.265i
Analytic cond. 1.038051.03805
Root an. cond. 1.018841.01884
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  + (−0.866 + 0.5i)2-s + (−0.366 − 0.633i)3-s + (0.499 − 0.866i)4-s + i·5-s + (0.633 + 0.366i)6-s + (2.59 + 1.5i)7-s + 0.999i·8-s + (1.23 − 2.13i)9-s + (−0.5 − 0.866i)10-s + (2.59 − 1.5i)11-s − 0.732·12-s + (3.5 + 0.866i)13-s − 3·14-s + (0.633 − 0.366i)15-s + (−0.5 − 0.866i)16-s + (−4.09 + 7.09i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.211 − 0.366i)3-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (0.258 + 0.149i)6-s + (0.981 + 0.566i)7-s + 0.353i·8-s + (0.410 − 0.711i)9-s + (−0.158 − 0.273i)10-s + (0.783 − 0.452i)11-s − 0.211·12-s + (0.970 + 0.240i)13-s − 0.801·14-s + (0.163 − 0.0945i)15-s + (−0.125 − 0.216i)16-s + (−0.993 + 1.72i)17-s + ⋯

Functional equation

Λ(s)=(130s/2 ΓC(s) L(s)=((0.964−0.265i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(130s/2 ΓC(s+1/2) L(s)=((0.964−0.265i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 130130    =    2⋅5⋅132 \cdot 5 \cdot 13
Sign: 0.964−0.265i0.964 - 0.265i
Analytic conductor: 1.038051.03805
Root analytic conductor: 1.018841.01884
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ130(101,⋅)\chi_{130} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 130, ( :1/2), 0.964−0.265i)(2,\ 130,\ (\ :1/2),\ 0.964 - 0.265i)

Particular Values

L(1)L(1) ≈\approx 0.866281+0.116874i0.866281 + 0.116874i
L(12)L(\frac12) ≈\approx 0.866281+0.116874i0.866281 + 0.116874i
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)
bad2 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
5 1−iT 1 - iT
13 1+(−3.5−0.866i)T 1 + (-3.5 - 0.866i)T
good3 1+(0.366+0.633i)T+(−1.5+2.59i)T2 1 + (0.366 + 0.633i)T + (-1.5 + 2.59i)T^{2}
7 1+(−2.59−1.5i)T+(3.5+6.06i)T2 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2}
11 1+(−2.59+1.5i)T+(5.5−9.52i)T2 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2}
17 1+(4.09−7.09i)T+(−8.5−14.7i)T2 1 + (4.09 - 7.09i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.401+0.232i)T+(9.5+16.4i)T2 1 + (0.401 + 0.232i)T + (9.5 + 16.4i)T^{2}
23 1+(4.73+8.19i)T+(−11.5+19.9i)T2 1 + (4.73 + 8.19i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.26−2.19i)T+(−14.5+25.1i)T2 1 + (-1.26 - 2.19i)T + (-14.5 + 25.1i)T^{2}
31 1+4.73iT−31T2 1 + 4.73iT - 31T^{2}
37 1+(0.696−0.401i)T+(18.5−32.0i)T2 1 + (0.696 - 0.401i)T + (18.5 - 32.0i)T^{2}
41 1+(9−5.19i)T+(20.5−35.5i)T2 1 + (9 - 5.19i)T + (20.5 - 35.5i)T^{2}
43 1+(1−1.73i)T+(−21.5−37.2i)T2 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2}
47 1+3iT−47T2 1 + 3iT - 47T^{2}
53 1−0.464T+53T2 1 - 0.464T + 53T^{2}
59 1+(−9−5.19i)T+(29.5+51.0i)T2 1 + (-9 - 5.19i)T + (29.5 + 51.0i)T^{2}
61 1+(3.09−5.36i)T+(−30.5−52.8i)T2 1 + (3.09 - 5.36i)T + (-30.5 - 52.8i)T^{2}
67 1+(33.5−58.0i)T2 1 + (33.5 - 58.0i)T^{2}
71 1+(5.19+3i)T+(35.5+61.4i)T2 1 + (5.19 + 3i)T + (35.5 + 61.4i)T^{2}
73 1+11.6iT−73T2 1 + 11.6iT - 73T^{2}
79 1+4.19T+79T2 1 + 4.19T + 79T^{2}
83 1−8.19iT−83T2 1 - 8.19iT - 83T^{2}
89 1+(−5.89+3.40i)T+(44.5−77.0i)T2 1 + (-5.89 + 3.40i)T + (44.5 - 77.0i)T^{2}
97 1+(−7.90−4.56i)T+(48.5+84.0i)T2 1 + (-7.90 - 4.56i)T + (48.5 + 84.0i)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

−13.41087801307522123354666433413, −12.10270905431703890984230128258, −11.26924998963423644375373818350, −10.32401655821139050957277387053, −8.811301885522161806246933870043, −8.255465819569279081369416427980, −6.61392585311012895962959764281, −6.10361021065949761062404583891, −4.11243522165841246154540440590, −1.73049891669617410671528272357, 1.64170616099128021979535656547, 4.01592167938776934834061317335, 5.13314679161140245856225262785, 7.02761420502294060023201411231, 8.052891047627317324138043260733, 9.177437026845674059168669774180, 10.18823552287621849241617557095, 11.24143945034493389253295960369, 11.81272724583241790205629408988, 13.35431885580636845362666310962

Graph of the ZZ-function along the critical line