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Properties

Label 2-275-55.54-c2-0-19
Degree $2$
Conductor $275$
Sign $-0.298 + 0.954i$
Analytic cond. $7.49320$
Root an. cond. $2.73737$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.54·2-s − 4.06i·3-s − 1.60·4-s + 6.29i·6-s + 8.53·7-s + 8.67·8-s − 7.53·9-s + (7.63 − 7.91i)11-s + 6.53i·12-s + 16.6·13-s − 13.2·14-s − 6.99·16-s − 5.33·17-s + 11.6·18-s − 0.884i·19-s + ⋯
L(s)  = 1  − 0.773·2-s − 1.35i·3-s − 0.401·4-s + 1.04i·6-s + 1.21·7-s + 1.08·8-s − 0.836·9-s + (0.694 − 0.719i)11-s + 0.544i·12-s + 1.28·13-s − 0.943·14-s − 0.437·16-s − 0.314·17-s + 0.647·18-s − 0.0465i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.298 + 0.954i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ -0.298 + 0.954i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.664343 - 0.904310i\)
\(L(\frac12)\) \(\approx\) \(0.664343 - 0.904310i\)
\(L(2)\) 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 \( 1 + (-7.63 + 7.91i)T \)
good2 \( 1 + 1.54T + 4T^{2} \)
3 \( 1 + 4.06iT - 9T^{2} \)
7 \( 1 - 8.53T + 49T^{2} \)
13 \( 1 - 16.6T + 169T^{2} \)
17 \( 1 + 5.33T + 289T^{2} \)
19 \( 1 + 0.884iT - 361T^{2} \)
23 \( 1 + 20.2iT - 529T^{2} \)
29 \( 1 - 40.3iT - 841T^{2} \)
31 \( 1 + 51.5T + 961T^{2} \)
37 \( 1 - 10.7iT - 1.36e3T^{2} \)
41 \( 1 + 36.1iT - 1.68e3T^{2} \)
43 \( 1 - 74.3T + 1.84e3T^{2} \)
47 \( 1 + 72.2iT - 2.20e3T^{2} \)
53 \( 1 - 13.8iT - 2.80e3T^{2} \)
59 \( 1 - 26.9T + 3.48e3T^{2} \)
61 \( 1 - 45.5iT - 3.72e3T^{2} \)
67 \( 1 + 78.3iT - 4.48e3T^{2} \)
71 \( 1 - 66.9T + 5.04e3T^{2} \)
73 \( 1 + 27.9T + 5.32e3T^{2} \)
79 \( 1 - 10.4iT - 6.24e3T^{2} \)
83 \( 1 + 121.T + 6.88e3T^{2} \)
89 \( 1 + 19.7T + 7.92e3T^{2} \)
97 \( 1 - 148. iT - 9.40e3T^{2} \)
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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

−11.21329017295191458897870703327, −10.68412121507869534329189182217, −8.936282543156973621050159755263, −8.570480906207678611881226423443, −7.65656981688318058336807066439, −6.73208515525696516077450806326, −5.46164743531322551427521228051, −3.97928976730334094731172215569, −1.79663384202373047193280664526, −0.893066240725227967800189039900, 1.51972990797870354710502042638, 3.93775327734774168703792532114, 4.48848989429657297572395247511, 5.65696638220545266910994232693, 7.44002469654889949771867253901, 8.419128542778070865990060709744, 9.248182120240452045444693542305, 9.822087517635428095260439861477, 11.00687618753789214377355658975, 11.26407923728512071367295203830

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