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Properties

Label 2-275-11.5-c1-0-14
Degree $2$
Conductor $275$
Sign $-0.706 + 0.707i$
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  + (−0.0430 − 0.132i)2-s + (−2.41 − 1.75i)3-s + (1.60 − 1.16i)4-s + (−0.128 + 0.395i)6-s + (2.88 − 2.09i)7-s + (−0.448 − 0.326i)8-s + (1.82 + 5.62i)9-s + (−2.45 − 2.23i)11-s − 5.91·12-s + (−0.369 − 1.13i)13-s + (−0.402 − 0.292i)14-s + (1.20 − 3.69i)16-s + (−1.67 + 5.16i)17-s + (0.667 − 0.484i)18-s + (−3.88 − 2.82i)19-s + ⋯
L(s)  = 1  + (−0.0304 − 0.0937i)2-s + (−1.39 − 1.01i)3-s + (0.801 − 0.582i)4-s + (−0.0525 + 0.161i)6-s + (1.09 − 0.793i)7-s + (−0.158 − 0.115i)8-s + (0.609 + 1.87i)9-s + (−0.739 − 0.673i)11-s − 1.70·12-s + (−0.102 − 0.315i)13-s + (−0.107 − 0.0781i)14-s + (0.300 − 0.923i)16-s + (−0.407 + 1.25i)17-s + (0.157 − 0.114i)18-s + (−0.891 − 0.647i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.706 + 0.707i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -0.706 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.354427 - 0.854143i\)
\(L(\frac12)\) \(\approx\) \(0.354427 - 0.854143i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.45 + 2.23i)T \)
good2 \( 1 + (0.0430 + 0.132i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (2.41 + 1.75i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-2.88 + 2.09i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.369 + 1.13i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.67 - 5.16i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.88 + 2.82i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.82T + 23T^{2} \)
29 \( 1 + (-3.35 + 2.43i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.397 - 1.22i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.55 + 1.85i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.18 - 3.76i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.23T + 43T^{2} \)
47 \( 1 + (-5.68 - 4.13i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.34 + 10.2i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.24 + 2.36i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.47 - 13.7i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 8.14T + 67T^{2} \)
71 \( 1 + (-2.63 + 8.09i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.90 + 5.02i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.121 - 0.374i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.483 - 1.48i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6.90T + 89T^{2} \)
97 \( 1 + (2.87 + 8.85i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.19199507257438282348534952490, −10.98858501797301376355275577583, −10.28359035076519542365128410129, −8.222042974280821192178128144469, −7.44546083580088207593851430950, −6.42135776254936968612130574661, −5.73225043328449311051572803323, −4.61049242530552996899742288382, −2.14815175105033771145995470227, −0.844433122191029738033225810004, 2.34830264848119762990122343904, 4.25887677183338904161677980145, 5.13592911622167621003929017969, 6.08230067644997316544757477311, 7.22492602627475390054345227434, 8.380843127316104384467064793006, 9.610181285683244331592384703288, 10.72225118341297362758690272844, 11.22955440588716620763582646318, 12.06144613170645387594144752991

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