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Properties

Label 1-6145-6145.272-r0-0-0
Degree $1$
Conductor $6145$
Sign $-0.997 - 0.0642i$
Analytic cond. $28.5372$
Root an. cond. $28.5372$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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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

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

Invariants

Degree: \(1\)
Conductor: \(6145\)    =    \(5 \cdot 1229\)
Sign: $-0.997 - 0.0642i$
Analytic conductor: \(28.5372\)
Root analytic conductor: \(28.5372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6145} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6145,\ (0:\ ),\ -0.997 - 0.0642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1022145625 - 3.177078293i\)
\(L(\frac12)\) \(\approx\) \(0.1022145625 - 3.177078293i\)
\(L(1)\) \(\approx\) \(1.559384943 - 1.242526840i\)
\(L(1)\) \(\approx\) \(1.559384943 - 1.242526840i\)

Euler product

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

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 $Z$-function along the critical line