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Properties

Label 1-275-275.158-r1-0-0
Degree $1$
Conductor $275$
Sign $-0.390 + 0.920i$
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  + (−0.587 + 0.809i)2-s + (−0.587 − 0.809i)3-s + (−0.309 − 0.951i)4-s + 6-s + (0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s + (−0.309 + 0.951i)9-s + (−0.587 + 0.809i)12-s + (0.951 + 0.309i)13-s − 14-s + (−0.809 + 0.587i)16-s − i·17-s + (−0.587 − 0.809i)18-s + (−0.309 + 0.951i)19-s + (0.309 − 0.951i)21-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.587 − 0.809i)3-s + (−0.309 − 0.951i)4-s + 6-s + (0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s + (−0.309 + 0.951i)9-s + (−0.587 + 0.809i)12-s + (0.951 + 0.309i)13-s − 14-s + (−0.809 + 0.587i)16-s − i·17-s + (−0.587 − 0.809i)18-s + (−0.309 + 0.951i)19-s + (0.309 − 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.390 + 0.920i$
Analytic conductor: \(29.5528\)
Root analytic conductor: \(29.5528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (1:\ ),\ -0.390 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4385123203 + 0.6620445630i\)
\(L(\frac12)\) \(\approx\) \(0.4385123203 + 0.6620445630i\)
\(L(1)\) \(\approx\) \(0.6382945249 + 0.1879339398i\)
\(L(1)\) \(\approx\) \(0.6382945249 + 0.1879339398i\)

Euler product

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

−25.85124781569095273482903152592, −23.95218634487782873983910980880, −23.27527638620072404435286321473, −22.09265891709543173338822050628, −21.52537275544936740910613646672, −20.49814578905563247067816433308, −20.04380396990282742434945701326, −18.68475075416480799188274056373, −17.634645835255796911006018511669, −17.171389451105652617961493684009, −16.18406529500565782742120705011, −15.14304380092864353077701328356, −13.800169746828504770069993127604, −12.79151342697971168814502193400, −11.56319385743707580428564886382, −10.87326156383979185622148138391, −10.27797319221102773518514796886, −9.11069128201808349811864411649, −8.18850799431914412715898819090, −6.89426435010369711211148434017, −5.39461441353871084499861353740, −4.1470586536230584140439520295, −3.4900157355298920765390629810, −1.66196929503244949617270554829, −0.37766718920795599823659466177, 1.13315870711192308275270120317, 2.23392634731791659533898378897, 4.52392691368985487073711548165, 5.70656807989622508196028286944, 6.27901149977961035925082934689, 7.530162563113206428239655396686, 8.29088999324804830251873009644, 9.284485000714725639171794456605, 10.69885831337050548481478890388, 11.514001438080595871832485997700, 12.615316096397420498568920903608, 13.84411352320440074735570345838, 14.576090545396554835365276050276, 15.882359087839611753048237184322, 16.52387608988956556070829400608, 17.632828407419606962043093885213, 18.424108131088442114181186682432, 18.72944455240647098432188714359, 20.000965697884629635262539821509, 21.23842711256073838706956367797, 22.56506124912327209381477280041, 23.19715833138109232809892349320, 24.138749261819787400142161892019, 24.87495924467476621334349303299, 25.40847632704590731157699257444

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