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Properties

Label 2-275-5.4-c3-0-33
Degree $2$
Conductor $275$
Sign $-0.447 + 0.894i$
Analytic cond. $16.2255$
Root an. cond. $4.02809$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.732i·2-s − 5.92i·3-s + 7.46·4-s − 4.33·6-s + 16.9i·7-s − 11.3i·8-s − 8.14·9-s − 11·11-s − 44.2i·12-s − 74.6i·13-s + 12.3·14-s + 51.4·16-s − 82.7i·17-s + 5.96i·18-s + 67.9·19-s + ⋯
L(s)  = 1  − 0.258i·2-s − 1.14i·3-s + 0.933·4-s − 0.295·6-s + 0.914i·7-s − 0.500i·8-s − 0.301·9-s − 0.301·11-s − 1.06i·12-s − 1.59i·13-s + 0.236·14-s + 0.803·16-s − 1.18i·17-s + 0.0780i·18-s + 0.820·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(16.2255\)
Root analytic conductor: \(4.02809\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.205873724\)
\(L(\frac12)\) \(\approx\) \(2.205873724\)
\(L(\frac{5}{2})\) 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 \( 1 + 11T \)
good2 \( 1 + 0.732iT - 8T^{2} \)
3 \( 1 + 5.92iT - 27T^{2} \)
7 \( 1 - 16.9iT - 343T^{2} \)
13 \( 1 + 74.6iT - 2.19e3T^{2} \)
17 \( 1 + 82.7iT - 4.91e3T^{2} \)
19 \( 1 - 67.9T + 6.85e3T^{2} \)
23 \( 1 + 13.3iT - 1.21e4T^{2} \)
29 \( 1 + 168.T + 2.43e4T^{2} \)
31 \( 1 + 65.4T + 2.97e4T^{2} \)
37 \( 1 - 40.8iT - 5.06e4T^{2} \)
41 \( 1 - 274.T + 6.89e4T^{2} \)
43 \( 1 - 2.28iT - 7.95e4T^{2} \)
47 \( 1 - 71.8iT - 1.03e5T^{2} \)
53 \( 1 - 149. iT - 1.48e5T^{2} \)
59 \( 1 + 545.T + 2.05e5T^{2} \)
61 \( 1 - 101.T + 2.26e5T^{2} \)
67 \( 1 - 411. iT - 3.00e5T^{2} \)
71 \( 1 + 470.T + 3.57e5T^{2} \)
73 \( 1 + 610. iT - 3.89e5T^{2} \)
79 \( 1 - 978.T + 4.93e5T^{2} \)
83 \( 1 + 26.1iT - 5.71e5T^{2} \)
89 \( 1 - 352.T + 7.04e5T^{2} \)
97 \( 1 - 847. iT - 9.12e5T^{2} \)
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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

−11.39187720530304607852978666658, −10.37729004847900980416997166833, −9.246080588196807681720515248037, −7.78724059047881095123096835639, −7.42288124505880634011388211004, −6.17606190625744771821137522576, −5.35479056437784461504669182192, −3.11068212365901905711739257995, −2.25284509230590881466768617636, −0.851229721931016078657904539697, 1.74258435986295856006686620945, 3.52235338251476367130621421762, 4.40532914784115481577302433263, 5.71474725150857950411829879500, 6.89592836435243531860983383236, 7.69446751185059908294898000242, 9.077351134164504131051939554741, 9.987869764390835244118762782425, 10.82889226810953119113800218340, 11.37258656348431889963782316086

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