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Properties

Label 2-364-364.55-c1-0-12
Degree 22
Conductor 364364
Sign −0.651−0.758i-0.651 - 0.758i
Analytic cond. 2.906552.90655
Root an. cond. 1.704861.70486
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  + (−0.895 + 1.09i)2-s + (1.32 + 2.29i)3-s + (−0.395 − 1.96i)4-s − i·5-s + (−3.69 − 0.604i)6-s + (2.29 + 1.32i)7-s + (2.49 + 1.32i)8-s + (−2 + 3.46i)9-s + (1.09 + 0.895i)10-s + (3.96 − 3.49i)12-s + (−2.59 + 2.5i)13-s + (−3.49 + 1.32i)14-s + (2.29 − 1.32i)15-s + (−3.68 + 1.55i)16-s + (3.46 + 2i)17-s + (−1.99 − 5.29i)18-s + ⋯
L(s)  = 1  + (−0.633 + 0.773i)2-s + (0.763 + 1.32i)3-s + (−0.197 − 0.980i)4-s − 0.447i·5-s + (−1.50 − 0.246i)6-s + (0.866 + 0.499i)7-s + (0.883 + 0.467i)8-s + (−0.666 + 1.15i)9-s + (0.346 + 0.283i)10-s + (1.14 − 1.01i)12-s + (−0.720 + 0.693i)13-s + (−0.935 + 0.353i)14-s + (0.591 − 0.341i)15-s + (−0.921 + 0.387i)16-s + (0.840 + 0.485i)17-s + (−0.471 − 1.24i)18-s + ⋯

Functional equation

Λ(s)=(364s/2 ΓC(s) L(s)=((−0.651−0.758i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(364s/2 ΓC(s+1/2) L(s)=((−0.651−0.758i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 364364    =    22⋅7⋅132^{2} \cdot 7 \cdot 13
Sign: −0.651−0.758i-0.651 - 0.758i
Analytic conductor: 2.906552.90655
Root analytic conductor: 1.704861.70486
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ364(55,⋅)\chi_{364} (55, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 364, ( :1/2), −0.651−0.758i)(2,\ 364,\ (\ :1/2),\ -0.651 - 0.758i)

Particular Values

L(1)L(1) ≈\approx 0.535753+1.16681i0.535753 + 1.16681i
L(12)L(\frac12) ≈\approx 0.535753+1.16681i0.535753 + 1.16681i
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+(0.895−1.09i)T 1 + (0.895 - 1.09i)T
7 1+(−2.29−1.32i)T 1 + (-2.29 - 1.32i)T
13 1+(2.59−2.5i)T 1 + (2.59 - 2.5i)T
good3 1+(−1.32−2.29i)T+(−1.5+2.59i)T2 1 + (-1.32 - 2.29i)T + (-1.5 + 2.59i)T^{2}
5 1+iT−5T2 1 + iT - 5T^{2}
11 1+(5.5−9.52i)T2 1 + (5.5 - 9.52i)T^{2}
17 1+(−3.46−2i)T+(8.5+14.7i)T2 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2}
19 1+(2.64−4.58i)T+(−9.5−16.4i)T2 1 + (2.64 - 4.58i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.29+1.32i)T+(11.5−19.9i)T2 1 + (-2.29 + 1.32i)T + (11.5 - 19.9i)T^{2}
29 1+(4+6.92i)T+(−14.5+25.1i)T2 1 + (4 + 6.92i)T + (-14.5 + 25.1i)T^{2}
31 1+5.29T+31T2 1 + 5.29T + 31T^{2}
37 1+(2+3.46i)T+(−18.5+32.0i)T2 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2}
41 1+(−1.73+i)T+(20.5−35.5i)T2 1 + (-1.73 + i)T + (20.5 - 35.5i)T^{2}
43 1+(4.58+2.64i)T+(21.5+37.2i)T2 1 + (4.58 + 2.64i)T + (21.5 + 37.2i)T^{2}
47 1+47T2 1 + 47T^{2}
53 1−12T+53T2 1 - 12T + 53T^{2}
59 1+(−3.96+6.87i)T+(−29.5−51.0i)T2 1 + (-3.96 + 6.87i)T + (-29.5 - 51.0i)T^{2}
61 1+(0.866+0.5i)T+(30.5+52.8i)T2 1 + (0.866 + 0.5i)T + (30.5 + 52.8i)T^{2}
67 1+(−4.58+2.64i)T+(33.5−58.0i)T2 1 + (-4.58 + 2.64i)T + (33.5 - 58.0i)T^{2}
71 1+(2.29+1.32i)T+(35.5+61.4i)T2 1 + (2.29 + 1.32i)T + (35.5 + 61.4i)T^{2}
73 1−10iT−73T2 1 - 10iT - 73T^{2}
79 1+5.29iT−79T2 1 + 5.29iT - 79T^{2}
83 1+15.8T+83T2 1 + 15.8T + 83T^{2}
89 1+(−12.1+7i)T+(44.5−77.0i)T2 1 + (-12.1 + 7i)T + (44.5 - 77.0i)T^{2}
97 1+(12.1+7i)T+(48.5+84.0i)T2 1 + (12.1 + 7i)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

−11.42180610048890283325792608418, −10.41490456810268182670816466536, −9.734432401310432928587534122716, −8.849779599998073160209682922103, −8.387706154426143758811677830004, −7.36437196545003431162610087590, −5.75475778695229415010168222526, −4.90517566859179913132916526640, −3.98141717784742200464360186786, −2.03183250527285891168741080489, 1.09820828921063432562835813670, 2.37201366860471064872016034342, 3.32494784240088333673747369785, 5.01383766193584606159292121362, 7.13195149262260718434757404505, 7.26482277096407286438863798177, 8.291560863073082427760567637642, 9.052770554970227153527011323210, 10.30071346821393827917605279210, 11.08970099581568772037501425188

Graph of the ZZ-function along the critical line