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Properties

Label 2-275-275.54-c0-0-0
Degree $2$
Conductor $275$
Sign $-0.535 - 0.844i$
Analytic cond. $0.137242$
Root an. cond. $0.370463$
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  + (−1.11 + 1.53i)3-s + (0.809 + 0.587i)4-s − 5-s + (−0.809 − 2.48i)9-s + (−0.309 + 0.951i)11-s + (−1.80 + 0.587i)12-s + (1.11 − 1.53i)15-s + (0.309 + 0.951i)16-s + (−0.809 − 0.587i)20-s + (1.11 + 0.363i)23-s + 25-s + (2.92 + 0.951i)27-s + (0.5 − 0.363i)31-s + (−1.11 − 1.53i)33-s + (0.809 − 2.48i)36-s + ⋯
L(s)  = 1  + (−1.11 + 1.53i)3-s + (0.809 + 0.587i)4-s − 5-s + (−0.809 − 2.48i)9-s + (−0.309 + 0.951i)11-s + (−1.80 + 0.587i)12-s + (1.11 − 1.53i)15-s + (0.309 + 0.951i)16-s + (−0.809 − 0.587i)20-s + (1.11 + 0.363i)23-s + 25-s + (2.92 + 0.951i)27-s + (0.5 − 0.363i)31-s + (−1.11 − 1.53i)33-s + (0.809 − 2.48i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.535 - 0.844i$
Analytic conductor: \(0.137242\)
Root analytic conductor: \(0.370463\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :0),\ -0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5303016296\)
\(L(\frac12)\) \(\approx\) \(0.5303016296\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
11 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.00950685906280784672832071854, −11.45352028259975348524775059570, −10.75071429949538029631324324957, −9.906628375183255657309525374208, −8.728257178555347555373324388040, −7.44581999637289669967682702325, −6.48010699447798527719121530140, −5.12544559215229894878358651966, −4.21002786238343686488886918270, −3.18890145839701212334771397704, 1.04192150792095182946622505339, 2.77429850919053569887030266032, 5.02084235255320985230793362627, 6.00419017891436222778166362708, 6.85609324003470617068731805478, 7.55988408330358144837082908999, 8.523003629587493838334422156577, 10.52572652615436940772930314315, 11.13389080345624546267022067194, 11.72388561070963576534103985022

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