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Properties

Label 1-275-275.27-r1-0-0
Degree $1$
Conductor $275$
Sign $-0.650 + 0.759i$
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  + (−0.951 + 0.309i)2-s + (−0.951 − 0.309i)3-s + (0.809 − 0.587i)4-s + 6-s + (0.951 + 0.309i)7-s + (−0.587 + 0.809i)8-s + (0.809 + 0.587i)9-s + (−0.951 + 0.309i)12-s + (−0.587 + 0.809i)13-s − 14-s + (0.309 − 0.951i)16-s + i·17-s + (−0.951 − 0.309i)18-s + (0.809 + 0.587i)19-s + (−0.809 − 0.587i)21-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (−0.951 − 0.309i)3-s + (0.809 − 0.587i)4-s + 6-s + (0.951 + 0.309i)7-s + (−0.587 + 0.809i)8-s + (0.809 + 0.587i)9-s + (−0.951 + 0.309i)12-s + (−0.587 + 0.809i)13-s − 14-s + (0.309 − 0.951i)16-s + i·17-s + (−0.951 − 0.309i)18-s + (0.809 + 0.587i)19-s + (−0.809 − 0.587i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.650 + 0.759i$
Analytic conductor: \(29.5528\)
Root analytic conductor: \(29.5528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (1:\ ),\ -0.650 + 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2386396939 + 0.5185796204i\)
\(L(\frac12)\) \(\approx\) \(0.2386396939 + 0.5185796204i\)
\(L(1)\) \(\approx\) \(0.5355298662 + 0.1340791280i\)
\(L(1)\) \(\approx\) \(0.5355298662 + 0.1340791280i\)

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 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
13 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (-0.951 - 0.309i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 - iT \)
59 \( 1 - T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.587 - 0.809i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 - T \)
83 \( 1 - iT \)
89 \( 1 + (-0.309 + 0.951i)T \)
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

−25.069771638124274078744585580385, −24.366934057851896862994694871232, −23.36122538967456628072576990718, −22.181200408743048680128099471312, −21.45761935317393412979886787017, −20.4707277547971802270024658378, −19.76452669007051378099214910181, −18.21815345345819051283941807956, −17.89492219390954789688141966416, −17.08551795314787971709427152368, −16.073366955054358468210969343540, −15.356673669293595736824407649108, −13.940093081461180949170913928919, −12.393251000148924347806063949709, −11.75188555229534146657507509243, −10.82275167236672167814288353643, −10.11570875125206905908541333863, −9.06583692610740742890485365743, −7.731737342974117864873739021976, −7.01459360663993239410692330086, −5.58266147471683668607737823204, −4.53573581288291199526945678079, −3.005294307815394165510831530747, −1.40776234995703137057267429776, −0.30778896771146806482209453826, 1.25801612229075941496493735050, 2.1810604967578698886483371848, 4.46128450238346452680828303751, 5.61075693965736093359391126895, 6.46482714553045445017425728132, 7.60676362407846908171814659180, 8.36102320197935392574338114574, 9.767296936784223825901717788688, 10.579448921179235168164909046064, 11.727674680393378630861673111242, 12.07086881391948448951052280689, 13.814905129721792575834879129625, 14.894843852967788829007493547, 15.87102071626517138069530088502, 16.91451563231483399357556009286, 17.40652334863507514481416904491, 18.41679131927158374105986006712, 18.947209038665768656095018444113, 20.16386749415566776680167030270, 21.25938057706446232093006487340, 22.118201232138843202360853121657, 23.436710742470564149537325702271, 24.20778045019224625592866355041, 24.62382425769045703353556203499, 25.853154467844691451937086913125

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