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Properties

Label 2-275-55.49-c1-0-0
Degree $2$
Conductor $275$
Sign $-0.998 + 0.0615i$
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  + (−0.453 + 0.147i)2-s + (−0.189 + 0.261i)3-s + (−1.43 + 1.04i)4-s + (0.0476 − 0.146i)6-s + (−1.57 − 2.17i)7-s + (1.05 − 1.45i)8-s + (0.894 + 2.75i)9-s + (−2.79 + 1.79i)11-s − 0.572i·12-s + (−4.43 + 1.44i)13-s + (1.03 + 0.753i)14-s + (0.829 − 2.55i)16-s + (−4.39 − 1.42i)17-s + (−0.812 − 1.11i)18-s + (−3.51 − 2.55i)19-s + ⋯
L(s)  = 1  + (−0.320 + 0.104i)2-s + (−0.109 + 0.150i)3-s + (−0.716 + 0.520i)4-s + (0.0194 − 0.0598i)6-s + (−0.596 − 0.821i)7-s + (0.374 − 0.514i)8-s + (0.298 + 0.917i)9-s + (−0.841 + 0.540i)11-s − 0.165i·12-s + (−1.23 + 0.400i)13-s + (0.277 + 0.201i)14-s + (0.207 − 0.638i)16-s + (−1.06 − 0.346i)17-s + (−0.191 − 0.263i)18-s + (−0.805 − 0.585i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.998 + 0.0615i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -0.998 + 0.0615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00521276 - 0.169125i\)
\(L(\frac12)\) \(\approx\) \(0.00521276 - 0.169125i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.79 - 1.79i)T \)
good2 \( 1 + (0.453 - 0.147i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.189 - 0.261i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (1.57 + 2.17i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (4.43 - 1.44i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.39 + 1.42i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.51 + 2.55i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.77iT - 23T^{2} \)
29 \( 1 + (2.43 - 1.77i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.737 - 2.26i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.25 - 8.61i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.78 - 1.29i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.06iT - 43T^{2} \)
47 \( 1 + (-2.56 + 3.52i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (6.02 - 1.95i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.50 + 6.90i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.23 - 3.78i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 7.31iT - 67T^{2} \)
71 \( 1 + (-0.369 + 1.13i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.600 - 0.826i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.08 + 3.33i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (10.5 + 3.43i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 2.76T + 89T^{2} \)
97 \( 1 + (17.6 - 5.72i)T + (78.4 - 57.0i)T^{2} \)
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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

−12.72898701727230851778807152220, −11.27149704048850711857795929058, −10.20733822751534834532740624053, −9.704958172327002467360312420841, −8.509561122591675232994469757851, −7.49011888025050124368721210263, −6.82936718292916140806442551402, −4.84508741030392928102808833908, −4.37930845926051528981757401607, −2.59658801490981471964841031404, 0.13835601180493194171804971886, 2.36721587893701577865455391756, 4.06473490948653134887677667655, 5.45536961875672936774276004628, 6.19741361453451780586322477695, 7.63337638108994897474121310002, 8.777522611491232700560468635224, 9.477248866162941902996358491300, 10.27953839691033417902375308285, 11.31062344977453860692860556674

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