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Properties

Label 2-275-5.4-c3-0-6
Degree $2$
Conductor $275$
Sign $-0.447 + 0.894i$
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  + 2.73i·2-s + 7.92i·3-s + 0.535·4-s − 21.6·6-s + 3.07i·7-s + 23.3i·8-s − 35.8·9-s − 11·11-s + 4.24i·12-s − 5.35i·13-s − 8.39·14-s − 59.4·16-s − 41.2i·17-s − 97.9i·18-s − 139.·19-s + ⋯
L(s)  = 1  + 0.965i·2-s + 1.52i·3-s + 0.0669·4-s − 1.47·6-s + 0.165i·7-s + 1.03i·8-s − 1.32·9-s − 0.301·11-s + 0.102i·12-s − 0.114i·13-s − 0.160·14-s − 0.928·16-s − 0.588i·17-s − 1.28i·18-s − 1.68·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.447 + 0.894i$
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: \((2,\ 275,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.411692835\)
\(L(\frac12)\) \(\approx\) \(1.411692835\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.73iT - 8T^{2} \)
3 \( 1 - 7.92iT - 27T^{2} \)
7 \( 1 - 3.07iT - 343T^{2} \)
13 \( 1 + 5.35iT - 2.19e3T^{2} \)
17 \( 1 + 41.2iT - 4.91e3T^{2} \)
19 \( 1 + 139.T + 6.85e3T^{2} \)
23 \( 1 - 111. iT - 1.21e4T^{2} \)
29 \( 1 - 24.9T + 2.43e4T^{2} \)
31 \( 1 - 31.4T + 2.97e4T^{2} \)
37 \( 1 - 13.1iT - 5.06e4T^{2} \)
41 \( 1 - 261.T + 6.89e4T^{2} \)
43 \( 1 - 57.7iT - 7.95e4T^{2} \)
47 \( 1 + 343. iT - 1.03e5T^{2} \)
53 \( 1 - 342. iT - 1.48e5T^{2} \)
59 \( 1 + 88.3T + 2.05e5T^{2} \)
61 \( 1 - 738.T + 2.26e5T^{2} \)
67 \( 1 - 342. iT - 3.00e5T^{2} \)
71 \( 1 + 207.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 + 1.29e3T + 4.93e5T^{2} \)
83 \( 1 + 441. iT - 5.71e5T^{2} \)
89 \( 1 - 1.48e3T + 7.04e5T^{2} \)
97 \( 1 - 1.34e3iT - 9.12e5T^{2} \)
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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

−11.80149489799244784035021206805, −10.94558535827759543346730258278, −10.19800569759812552753653228330, −9.117106124860681849433193532415, −8.332680359209660022192099951077, −7.15713222415703252182524346391, −5.93541461466281903632609737641, −5.10930925873633380741585036787, −4.05790502153425892943474322216, −2.56289422630260238368321333213, 0.50684220798334231562226717325, 1.80197377956965645335416910917, 2.64246023915408329191107017449, 4.20654532501381803700741064982, 6.13244279567184014590871469775, 6.78465561241106532973337912711, 7.81772965597152635372328368448, 8.794735396771399739850685654308, 10.25068997804125113722272531850, 10.95629529726510118124544253037

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