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Properties

Label 2-275-55.54-c2-0-9
Degree $2$
Conductor $275$
Sign $0.297 - 0.954i$
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  − 3.15·2-s + 3.03i·3-s + 5.98·4-s − 9.57i·6-s − 5.67·7-s − 6.26·8-s − 0.187·9-s + (10.8 − 1.76i)11-s + 18.1i·12-s + 17.2·13-s + 17.9·14-s − 4.13·16-s + 3.43·17-s + 0.593·18-s − 18.5i·19-s + ⋯
L(s)  = 1  − 1.57·2-s + 1.01i·3-s + 1.49·4-s − 1.59i·6-s − 0.810·7-s − 0.783·8-s − 0.0208·9-s + (0.987 − 0.160i)11-s + 1.51i·12-s + 1.32·13-s + 1.27·14-s − 0.258·16-s + 0.202·17-s + 0.0329·18-s − 0.977i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.297 - 0.954i$
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: \((2,\ 275,\ (\ :1),\ 0.297 - 0.954i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.605540 + 0.445417i\)
\(L(\frac12)\) \(\approx\) \(0.605540 + 0.445417i\)
\(L(2)\) 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 \( 1 + (-10.8 + 1.76i)T \)
good2 \( 1 + 3.15T + 4T^{2} \)
3 \( 1 - 3.03iT - 9T^{2} \)
7 \( 1 + 5.67T + 49T^{2} \)
13 \( 1 - 17.2T + 169T^{2} \)
17 \( 1 - 3.43T + 289T^{2} \)
19 \( 1 + 18.5iT - 361T^{2} \)
23 \( 1 + 15.7iT - 529T^{2} \)
29 \( 1 - 52.6iT - 841T^{2} \)
31 \( 1 - 27.5T + 961T^{2} \)
37 \( 1 + 40.1iT - 1.36e3T^{2} \)
41 \( 1 - 74.8iT - 1.68e3T^{2} \)
43 \( 1 - 9.72T + 1.84e3T^{2} \)
47 \( 1 - 18.2iT - 2.20e3T^{2} \)
53 \( 1 - 75.2iT - 2.80e3T^{2} \)
59 \( 1 + 1.75T + 3.48e3T^{2} \)
61 \( 1 - 48.5iT - 3.72e3T^{2} \)
67 \( 1 + 35.4iT - 4.48e3T^{2} \)
71 \( 1 - 110.T + 5.04e3T^{2} \)
73 \( 1 + 50.7T + 5.32e3T^{2} \)
79 \( 1 - 12.7iT - 6.24e3T^{2} \)
83 \( 1 - 112.T + 6.88e3T^{2} \)
89 \( 1 + 66.3T + 7.92e3T^{2} \)
97 \( 1 - 65.1iT - 9.40e3T^{2} \)
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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

−11.18375712420115403221783557938, −10.72785325420579166517891410793, −9.734541026273748422804982442298, −9.151409690907126686549061135378, −8.492355200179713416782755410960, −7.06467105143445392365095158374, −6.24162392222516067652803504677, −4.46812758174378441224923041046, −3.19974411543740428147823811413, −1.13412254165529904664187492635, 0.831319493435122000350300049576, 1.91174932055751712797230484988, 3.77920323737906767393376850189, 6.17593528556523166993076453079, 6.70694812459188866359764852026, 7.77915996521602994239681534898, 8.499781872804150810421435284209, 9.575279385570908120387987283673, 10.18382829055922685182611807188, 11.44356627849496546716516762128

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