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Properties

Label 1-275-275.252-r0-0-0
Degree $1$
Conductor $275$
Sign $-0.992 - 0.125i$
Analytic cond. $1.27709$
Root an. cond. $1.27709$
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  + (−0.951 − 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)6-s − i·7-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (−0.951 + 0.309i)12-s + (−0.951 + 0.309i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + i·18-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)21-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)6-s − i·7-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (−0.951 + 0.309i)12-s + (−0.951 + 0.309i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + i·18-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.992 - 0.125i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (252, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ -0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0003946186574 + 0.006272283947i\)
\(L(\frac12)\) \(\approx\) \(0.0003946186574 + 0.006272283947i\)
\(L(1)\) \(\approx\) \(0.4203195938 + 0.02121785476i\)
\(L(1)\) \(\approx\) \(0.4203195938 + 0.02121785476i\)

Euler product

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

−25.27241640095655496869230525088, −24.36606825811299332182794616288, −23.81915616016124878980155381390, −22.572877044018945343562369739243, −21.69328735763596389074141063912, −20.32586547600881190819395732164, −19.36238631531542896203365029624, −18.65428200143757117020993779326, −17.9299555786641348556981164002, −17.11053874487810949105562306205, −16.21666113790301862311427078424, −15.19972391331669587610215320004, −14.20506969417960488809275350108, −12.71402016987440952405939004478, −11.930299581049564234462994480627, −11.0902074144043419754319794000, −9.90349898417007641180715106263, −8.866492616653294279180286952047, −7.79695231876337845263088164946, −7.00492461861717193602223652397, −5.86637150645770304612044859078, −5.15007826611004775430546419415, −2.69362400825482576562922640918, −1.76412452684449960174486469787, −0.005893687424091163040010494, 1.73832263793082879580991484378, 3.45747872377958357037487859561, 4.33236061798387083607845861445, 5.8976792876578029081984806465, 6.99502462671681516720472230213, 8.069884207777068892072120223295, 9.29498868761753209182329132295, 10.256971820951947434894479020276, 10.6504445791718656972719637271, 11.85649644187782735990621544105, 12.64203743125256351460278804388, 14.30939645351800637726709768165, 15.2803369351107844687245921514, 16.480512764442379709282784816485, 16.92115363623949106613223878999, 17.636923070580969927657927260382, 18.861264582659253528013571210567, 19.82616196142103582075983934393, 20.64837806152432637125876725434, 21.449527955539938919663013203435, 22.315686130285293253672581804970, 23.47053058187815999360656707860, 24.310940690599545452074442502533, 25.72616235380946150370113817365, 26.360664379233682445248892567596

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