Properties

Label 1-6145-6145.272-r0-0-0
Degree 11
Conductor 61456145
Sign 0.9970.0642i-0.997 - 0.0642i
Analytic cond. 28.537228.5372
Root an. cond. 28.537228.5372
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.498i)2-s + (0.856 − 0.516i)3-s + (0.502 − 0.864i)4-s + (0.485 − 0.874i)6-s + (−0.213 + 0.976i)7-s + (0.00511 − 0.999i)8-s + (0.467 − 0.884i)9-s + (−0.971 + 0.238i)11-s + (−0.0153 − 0.999i)12-s + (−0.157 − 0.987i)13-s + (0.302 + 0.953i)14-s + (−0.494 − 0.869i)16-s + (0.336 − 0.941i)17-s + (−0.0358 − 0.999i)18-s + (0.985 − 0.168i)19-s + ⋯
L(s)  = 1  + (0.866 − 0.498i)2-s + (0.856 − 0.516i)3-s + (0.502 − 0.864i)4-s + (0.485 − 0.874i)6-s + (−0.213 + 0.976i)7-s + (0.00511 − 0.999i)8-s + (0.467 − 0.884i)9-s + (−0.971 + 0.238i)11-s + (−0.0153 − 0.999i)12-s + (−0.157 − 0.987i)13-s + (0.302 + 0.953i)14-s + (−0.494 − 0.869i)16-s + (0.336 − 0.941i)17-s + (−0.0358 − 0.999i)18-s + (0.985 − 0.168i)19-s + ⋯

Functional equation

Λ(s)=(6145s/2ΓR(s)L(s)=((0.9970.0642i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0642i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(6145s/2ΓR(s)L(s)=((0.9970.0642i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0642i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 61456145    =    512295 \cdot 1229
Sign: 0.9970.0642i-0.997 - 0.0642i
Analytic conductor: 28.537228.5372
Root analytic conductor: 28.537228.5372
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ6145(272,)\chi_{6145} (272, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 6145, (0: ), 0.9970.0642i)(1,\ 6145,\ (0:\ ),\ -0.997 - 0.0642i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.10221456253.177078293i0.1022145625 - 3.177078293i
L(12)L(\frac12) \approx 0.10221456253.177078293i0.1022145625 - 3.177078293i
L(1)L(1) \approx 1.5593849431.242526840i1.559384943 - 1.242526840i
L(1)L(1) \approx 1.5593849431.242526840i1.559384943 - 1.242526840i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
1229 1 1
good2 1+(0.8660.498i)T 1 + (0.866 - 0.498i)T
3 1+(0.8560.516i)T 1 + (0.856 - 0.516i)T
7 1+(0.213+0.976i)T 1 + (-0.213 + 0.976i)T
11 1+(0.971+0.238i)T 1 + (-0.971 + 0.238i)T
13 1+(0.1570.987i)T 1 + (-0.157 - 0.987i)T
17 1+(0.3360.941i)T 1 + (0.336 - 0.941i)T
19 1+(0.9850.168i)T 1 + (0.985 - 0.168i)T
23 1+(0.9130.407i)T 1 + (-0.913 - 0.407i)T
29 1+(0.8200.571i)T 1 + (-0.820 - 0.571i)T
31 1+(0.881+0.471i)T 1 + (0.881 + 0.471i)T
37 1+(0.705+0.708i)T 1 + (0.705 + 0.708i)T
41 1+(0.2280.973i)T 1 + (0.228 - 0.973i)T
43 1+(0.6000.799i)T 1 + (-0.600 - 0.799i)T
47 1+(0.625+0.780i)T 1 + (0.625 + 0.780i)T
53 1+(0.994+0.107i)T 1 + (-0.994 + 0.107i)T
59 1+(0.596+0.802i)T 1 + (-0.596 + 0.802i)T
61 1+(0.767+0.641i)T 1 + (0.767 + 0.641i)T
67 1+(0.8980.439i)T 1 + (-0.898 - 0.439i)T
71 1+(0.893+0.448i)T 1 + (0.893 + 0.448i)T
73 1+(0.3500.936i)T 1 + (-0.350 - 0.936i)T
79 1+(0.00511+0.999i)T 1 + (0.00511 + 0.999i)T
83 1+(0.6900.723i)T 1 + (0.690 - 0.723i)T
89 1+(0.605+0.796i)T 1 + (-0.605 + 0.796i)T
97 1+(0.7700.637i)T 1 + (-0.770 - 0.637i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−17.88529400674478663969203723470, −16.968378156575721369249372797578, −16.421025140167665080585039192091, −16.03129003495424016779840053664, −15.3499525435746415218594023657, −14.53509247460547810924746550654, −14.19447683215840649911011480733, −13.46070483675899442736650024819, −13.1382656665017760233091488249, −12.31412303095123801966933392283, −11.34435428020692138957083179753, −10.82550844355015419088928816896, −9.93386233956447705027919036334, −9.474411713374449727850720914425, −8.36052900861929293750452873445, −7.84815507420930728217084439204, −7.42251009380563753008368247822, −6.54824131635754690869441892536, −5.75185012639916288474715446895, −4.93926452478877414109335672461, −4.28720696530864709127676641045, −3.678154821530236051595460429025, −3.143746497370113858660870292320, −2.25936014112272806346690102291, −1.45983415131121901928031730520, 0.439907727126570573750693789936, 1.457829173503341991604247321054, 2.42938361343367680079705053775, 2.75625819458420984032811573639, 3.29898113233700086012727134438, 4.299686039398763112160646896003, 5.19117614504858059173610013031, 5.678413920665602396310808301214, 6.44067659524957542549926965974, 7.37969017238794702022078820516, 7.834546081449273187586002883757, 8.692675790950234571410824518811, 9.67144698159782050862714906165, 9.86789793687664400970590153734, 10.818703202561821597208487892135, 11.78615308658147139618921899789, 12.264064324833066269384098913165, 12.69281197234454763605260920638, 13.59582881428072864690166236932, 13.76939331996190308554137292920, 14.7209374723822797818783012540, 15.333940085733216498591521451765, 15.6509794029213124417927382035, 16.334483780611033855507838123476, 17.77329342501459707476914274893

Graph of the ZZ-function along the critical line