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Properties

Label 2-275-55.54-c2-0-24
Degree $2$
Conductor $275$
Sign $-0.991 + 0.126i$
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  − 3.88·2-s − 4.12i·3-s + 11.1·4-s + 16.0i·6-s + 3.44·7-s − 27.5·8-s − 7.98·9-s + (−6.12 − 9.13i)11-s − 45.7i·12-s + 6.33·13-s − 13.3·14-s + 62.8·16-s + 0.715·17-s + 31.0·18-s − 20.9i·19-s + ⋯
L(s)  = 1  − 1.94·2-s − 1.37i·3-s + 2.77·4-s + 2.66i·6-s + 0.491·7-s − 3.44·8-s − 0.887·9-s + (−0.557 − 0.830i)11-s − 3.81i·12-s + 0.487·13-s − 0.955·14-s + 3.92·16-s + 0.0421·17-s + 1.72·18-s − 1.10i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.991 + 0.126i$
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: \((2,\ 275,\ (\ :1),\ -0.991 + 0.126i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0312494 - 0.490359i\)
\(L(\frac12)\) \(\approx\) \(0.0312494 - 0.490359i\)
\(L(2)\) 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 \( 1 + (6.12 + 9.13i)T \)
good2 \( 1 + 3.88T + 4T^{2} \)
3 \( 1 + 4.12iT - 9T^{2} \)
7 \( 1 - 3.44T + 49T^{2} \)
13 \( 1 - 6.33T + 169T^{2} \)
17 \( 1 - 0.715T + 289T^{2} \)
19 \( 1 + 20.9iT - 361T^{2} \)
23 \( 1 + 4.55iT - 529T^{2} \)
29 \( 1 - 0.262iT - 841T^{2} \)
31 \( 1 + 9.37T + 961T^{2} \)
37 \( 1 + 19.4iT - 1.36e3T^{2} \)
41 \( 1 + 24.1iT - 1.68e3T^{2} \)
43 \( 1 + 63.5T + 1.84e3T^{2} \)
47 \( 1 + 34.9iT - 2.20e3T^{2} \)
53 \( 1 - 60.7iT - 2.80e3T^{2} \)
59 \( 1 + 47.3T + 3.48e3T^{2} \)
61 \( 1 - 108. iT - 3.72e3T^{2} \)
67 \( 1 - 96.1iT - 4.48e3T^{2} \)
71 \( 1 - 20.7T + 5.04e3T^{2} \)
73 \( 1 - 98.4T + 5.32e3T^{2} \)
79 \( 1 + 114. iT - 6.24e3T^{2} \)
83 \( 1 + 127.T + 6.88e3T^{2} \)
89 \( 1 - 3.05T + 7.92e3T^{2} \)
97 \( 1 + 91.6iT - 9.40e3T^{2} \)
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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

−11.15099986974829306126170937801, −10.26391993180568343284144878138, −8.919879777826456006465057847842, −8.338708521837625040950345529367, −7.50603592107046715735712032167, −6.77953475347070688201114754176, −5.77283213446028368054941340153, −2.81223318387262606144342496414, −1.63028407885536539352860776361, −0.46210047488348374012935921645, 1.75273253737054830151589125069, 3.40053642919065332126532007390, 5.07390091282283615550359752850, 6.46092690698476670986689074351, 7.77153934300612235131534074983, 8.418544132672706656173814501201, 9.572671598173847842053578562507, 9.944365543720813004219594980837, 10.80444061845949768462308669510, 11.42665033541306573641485303589

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