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Properties

Label 1-275-275.256-r0-0-0
Degree $1$
Conductor $275$
Sign $-0.555 - 0.831i$
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  + (0.309 + 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s + (−0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)12-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.809 + 0.587i)17-s + (−0.809 − 0.587i)18-s + (−0.809 − 0.587i)19-s + (0.309 − 0.951i)21-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s + (−0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)12-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.809 + 0.587i)17-s + (−0.809 − 0.587i)18-s + (−0.809 − 0.587i)19-s + (0.309 − 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ -0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2915663506 + 0.5456213693i\)
\(L(\frac12)\) \(\approx\) \(-0.2915663506 + 0.5456213693i\)
\(L(1)\) \(\approx\) \(0.4914389775 + 0.6811024740i\)
\(L(1)\) \(\approx\) \(0.4914389775 + 0.6811024740i\)

Euler product

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

−24.838462637550471026199045654584, −24.266996919526351393536803242960, −22.92171442391796175346557665401, −22.568519391528593718869460503910, −21.41813790091548156265302164483, −20.2979429475024263005043210334, −19.622578754828031374459321240523, −18.87492063360750005658155776800, −18.116914852538958577366279489179, −17.147423844898763514813087444711, −15.54230779555828871993282115403, −14.557911858094311216173539198237, −13.63379681036814523625417977906, −12.63322221731739915596227506067, −12.29954180611424292926065173292, −11.11191476090845570422726564664, −9.8475648009711943241525532956, −8.99721350350273093411465023703, −7.91059786652450821837479348263, −6.470927377805690672710192826093, −5.605334068181249752979741904285, −4.08334081895156750598310203066, −2.73749117736127545267272466966, −2.15517244577418359082884956182, −0.3388842026575469932206501523, 2.699231489354010604572122660079, 3.97857530170128970861945793017, 4.56792473928735766372612917276, 5.924520218879733203684160709220, 6.91309592005146557315890515130, 8.067414847959154816259715520981, 9.1796499763857846147173464366, 9.84040923837361569182408891093, 11.06804550813361798799888351103, 12.540877872612240904971866622191, 13.52934467948447138836164292919, 14.347248384662018473675070753818, 15.299530171143764280274540420697, 15.998466092763768576918870254907, 16.91102215844034220926880308836, 17.49892469733216190622620741996, 19.128942706002115337108023609985, 19.86615637178849572403885149692, 21.11005843273404618068443265569, 22.01013603956624056519844561545, 22.50565457532914772250196637025, 23.63235367181071032876739245203, 24.41275924075764774150136738503, 25.79792785368763267073092874278, 26.01169582133581217313392265817

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