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Properties

Label 2-275-55.14-c1-0-6
Degree $2$
Conductor $275$
Sign $0.605 - 0.795i$
Analytic cond. $2.19588$
Root an. cond. $1.48185$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.796 + 1.09i)2-s + (−0.547 + 0.177i)3-s + (0.0501 − 0.154i)4-s + (−0.631 − 0.458i)6-s + (3.47 + 1.12i)7-s + (2.78 − 0.905i)8-s + (−2.15 + 1.56i)9-s + (0.490 − 3.28i)11-s + 0.0933i·12-s + (1.66 + 2.29i)13-s + (1.52 + 4.70i)14-s + (2.95 + 2.14i)16-s + (−2.17 + 2.98i)17-s + (−3.44 − 1.11i)18-s + (0.0293 + 0.0904i)19-s + ⋯
L(s)  = 1  + (0.563 + 0.775i)2-s + (−0.315 + 0.102i)3-s + (0.0250 − 0.0771i)4-s + (−0.257 − 0.187i)6-s + (1.31 + 0.426i)7-s + (0.985 − 0.320i)8-s + (−0.719 + 0.522i)9-s + (0.147 − 0.989i)11-s + 0.0269i·12-s + (0.461 + 0.635i)13-s + (0.408 + 1.25i)14-s + (0.738 + 0.536i)16-s + (−0.526 + 0.724i)17-s + (−0.811 − 0.263i)18-s + (0.00674 + 0.0207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58132 + 0.783282i\)
\(L(\frac12)\) \(\approx\) \(1.58132 + 0.783282i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + (-0.490 + 3.28i)T \)
good2 \( 1 + (-0.796 - 1.09i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.547 - 0.177i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-3.47 - 1.12i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.66 - 2.29i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.17 - 2.98i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.0293 - 0.0904i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.16iT - 23T^{2} \)
29 \( 1 + (-2.08 + 6.42i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.48 - 3.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (9.35 + 3.04i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.57 + 7.91i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.96iT - 43T^{2} \)
47 \( 1 + (2.11 - 0.687i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.75 + 2.42i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.62 + 8.09i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.86 - 4.98i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 13.4iT - 67T^{2} \)
71 \( 1 + (6.71 + 4.88i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.25 + 0.407i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (11.2 - 8.15i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.25 - 8.61i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (2.54 + 3.50i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.82581622237474609562402579723, −11.13515537568771760632097369027, −10.49094728463797895375319356953, −8.763666927727989512659430628808, −8.183802937608120911519728877912, −6.87936154095839316225680060626, −5.76309732291020940384423621860, −5.23083623833182363753134763406, −4.01598881285734240787012984306, −1.86136518274029525010742001933, 1.62070469238405596816837411239, 3.13660324950560815314500896310, 4.46186276529954148275460749834, 5.25874087800081839256427862937, 6.86866069045891888628231317847, 7.83933199515172395169052397458, 8.839713753299467803974396506777, 10.30576940478997341498097978742, 11.14169133486263777075878871444, 11.68136184301286069340428938670

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