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Properties

Label 1-275-275.156-r1-0-0
Degree $1$
Conductor $275$
Sign $0.269 + 0.962i$
Analytic cond. $29.5528$
Root an. cond. $29.5528$
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  − 2-s + (−0.809 − 0.587i)3-s + 4-s + (0.809 + 0.587i)6-s + (−0.309 − 0.951i)7-s − 8-s + (0.309 + 0.951i)9-s + (−0.809 − 0.587i)12-s + (−0.309 − 0.951i)13-s + (0.309 + 0.951i)14-s + 16-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)18-s − 19-s + (−0.309 + 0.951i)21-s + ⋯
L(s)  = 1  − 2-s + (−0.809 − 0.587i)3-s + 4-s + (0.809 + 0.587i)6-s + (−0.309 − 0.951i)7-s − 8-s + (0.309 + 0.951i)9-s + (−0.809 − 0.587i)12-s + (−0.309 − 0.951i)13-s + (0.309 + 0.951i)14-s + 16-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)18-s − 19-s + (−0.309 + 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.269 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.269 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.269 + 0.962i$
Analytic conductor: \(29.5528\)
Root analytic conductor: \(29.5528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (1:\ ),\ 0.269 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1627700232 + 0.1234155951i\)
\(L(\frac12)\) \(\approx\) \(0.1627700232 + 0.1234155951i\)
\(L(1)\) \(\approx\) \(0.4288794370 - 0.1202270447i\)
\(L(1)\) \(\approx\) \(0.4288794370 - 0.1202270447i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 - T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−25.51275634669511942461246200407, −24.45321860370052595947139160610, −23.580874865557935305936021658724, −22.398880514061954225642311796574, −21.47054313530937074042323348968, −20.885531410523523172087865215798, −19.57655142855930279812619460502, −18.63815170321932394990105034159, −18.016466209928838248064626254169, −16.79006456890355096949009673164, −16.36458527129575169467979583815, −15.37534178924367234075545559669, −14.53148367711249756381284382881, −12.55119224864643262213680516601, −11.879901613978863027558893627262, −11.03079475327269748211589207482, −9.881867456916361842447632357457, −9.31215142525860988761586652962, −8.19441814149989709892018188083, −6.727540365740421499222588340462, −6.043051378405690558681716931091, −4.79918447690241047029324513101, −3.21489525804341510140332550922, −1.81480319684089259854038666933, −0.125198895715001529712040911907, 0.88255858076963460476281397734, 2.14054717064523425130273782129, 3.77712682326646500780380984915, 5.54677667204757809171860434850, 6.42992161331131297995496749272, 7.51214189018766064530610199188, 8.07371732677213652915456660431, 9.74295669750100630252226350740, 10.47374184885482698033321080890, 11.26804905024961687673779905309, 12.422485272250929010734514910903, 13.1750043612981563400083706149, 14.6510293619658589380321398200, 15.85390884153535939482958610575, 16.852529451307600093188389281814, 17.27131136471120892814679946184, 18.204103947607178107591836502053, 19.19640251208663051213901382150, 19.82494046255374654751070931500, 20.90472182218750090082668912846, 22.069214432771567928942824444160, 23.19239355579902487879360355619, 23.8553381731238092985527857234, 24.82094619189348495658970222011, 25.70853577926830943466006313022

Graph of the $Z$-function along the critical line