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Properties

Label 2-275-5.4-c3-0-1
Degree $2$
Conductor $275$
Sign $0.894 + 0.447i$
Analytic cond. $16.2255$
Root an. cond. $4.02809$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.72i·2-s + 8.29i·3-s − 14.2·4-s − 39.1·6-s + 5.48i·7-s − 29.6i·8-s − 41.8·9-s − 11·11-s − 118. i·12-s − 84.9i·13-s − 25.8·14-s + 25.8·16-s + 119. i·17-s − 197. i·18-s + 54.4·19-s + ⋯
L(s)  = 1  + 1.66i·2-s + 1.59i·3-s − 1.78·4-s − 2.66·6-s + 0.296i·7-s − 1.31i·8-s − 1.54·9-s − 0.301·11-s − 2.85i·12-s − 1.81i·13-s − 0.494·14-s + 0.403·16-s + 1.70i·17-s − 2.58i·18-s + 0.657·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(16.2255\)
Root analytic conductor: \(4.02809\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6759077920\)
\(L(\frac12)\) \(\approx\) \(0.6759077920\)
\(L(\frac{5}{2})\) 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 \( 1 + 11T \)
good2 \( 1 - 4.72iT - 8T^{2} \)
3 \( 1 - 8.29iT - 27T^{2} \)
7 \( 1 - 5.48iT - 343T^{2} \)
13 \( 1 + 84.9iT - 2.19e3T^{2} \)
17 \( 1 - 119. iT - 4.91e3T^{2} \)
19 \( 1 - 54.4T + 6.85e3T^{2} \)
23 \( 1 + 83.8iT - 1.21e4T^{2} \)
29 \( 1 + 296.T + 2.43e4T^{2} \)
31 \( 1 + 18.7T + 2.97e4T^{2} \)
37 \( 1 - 188. iT - 5.06e4T^{2} \)
41 \( 1 + 209.T + 6.89e4T^{2} \)
43 \( 1 - 183. iT - 7.95e4T^{2} \)
47 \( 1 - 142. iT - 1.03e5T^{2} \)
53 \( 1 - 610. iT - 1.48e5T^{2} \)
59 \( 1 + 657.T + 2.05e5T^{2} \)
61 \( 1 - 171.T + 2.26e5T^{2} \)
67 \( 1 - 116. iT - 3.00e5T^{2} \)
71 \( 1 + 595.T + 3.57e5T^{2} \)
73 \( 1 + 629. iT - 3.89e5T^{2} \)
79 \( 1 - 935.T + 4.93e5T^{2} \)
83 \( 1 + 81.2iT - 5.71e5T^{2} \)
89 \( 1 - 245.T + 7.04e5T^{2} \)
97 \( 1 + 395. iT - 9.12e5T^{2} \)
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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

−12.57799916908343130269457437814, −10.92842778382003598494885953453, −10.24473706013545104077301983528, −9.257635684984383323043225627305, −8.389401812096744054446755482451, −7.64408028145699804128720535006, −5.99644043402791295616668810242, −5.48366245188070697856531022938, −4.48133110358615184364118860037, −3.29808537536948226357517800564, 0.26039580836544188718862138601, 1.54390704751961995439290636569, 2.36515163695079310186133518764, 3.72768400217465217086850630332, 5.22604095720427103494935853470, 6.86659324522835948518007764776, 7.51686081552272616642726848797, 9.002781477432539888605473244754, 9.617455278794651899312289930511, 11.06713279403074379276289708453

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