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Properties

Label 2-275-5.4-c1-0-3
Degree $2$
Conductor $275$
Sign $0.894 - 0.447i$
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  − 1.61i·2-s + 2.61i·3-s − 0.618·4-s + 4.23·6-s + 2.85i·7-s − 2.23i·8-s − 3.85·9-s + 11-s − 1.61i·12-s + 6.23i·13-s + 4.61·14-s − 4.85·16-s + 0.618i·17-s + 6.23i·18-s + 6.70·19-s + ⋯
L(s)  = 1  − 1.14i·2-s + 1.51i·3-s − 0.309·4-s + 1.72·6-s + 1.07i·7-s − 0.790i·8-s − 1.28·9-s + 0.301·11-s − 0.467i·12-s + 1.72i·13-s + 1.23·14-s − 1.21·16-s + 0.149i·17-s + 1.46i·18-s + 1.53·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31192 + 0.309704i\)
\(L(\frac12)\) \(\approx\) \(1.31192 + 0.309704i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + 1.61iT - 2T^{2} \)
3 \( 1 - 2.61iT - 3T^{2} \)
7 \( 1 - 2.85iT - 7T^{2} \)
13 \( 1 - 6.23iT - 13T^{2} \)
17 \( 1 - 0.618iT - 17T^{2} \)
19 \( 1 - 6.70T + 19T^{2} \)
23 \( 1 + 4.09iT - 23T^{2} \)
29 \( 1 - 1.38T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + 11.9iT - 47T^{2} \)
53 \( 1 - 9.32iT - 53T^{2} \)
59 \( 1 + 0.527T + 59T^{2} \)
61 \( 1 - 0.0901T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 8.18T + 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 + 5.85T + 79T^{2} \)
83 \( 1 + 10.1iT - 83T^{2} \)
89 \( 1 - 6.90T + 89T^{2} \)
97 \( 1 + 1.61iT - 97T^{2} \)
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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

−11.77313390733045625648710442125, −11.04803988551230728843729594545, −10.13695559219533997233265132906, −9.261460807052151635365222418003, −8.925008519717610174094785141781, −6.94351074381715114845564399556, −5.54306444368974639810309278758, −4.35981718380402661117255421022, −3.46041603517728675052092749729, −2.14508211554959182541817510430, 1.15977746197021019456222266649, 3.08813634051600965911902595190, 5.15287036188343262801988004778, 6.14296261620476902447397527127, 7.08360806371300621214211230447, 7.66437481099548310813439346033, 8.217450714693806831918879522427, 9.803087070522431268092524797561, 11.08411434513601382171811660585, 11.95085883658496813473780441732

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