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Properties

Label 2-275-5.3-c2-0-25
Degree $2$
Conductor $275$
Sign $-0.437 + 0.899i$
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  + (2.31 − 2.31i)2-s + (0.535 + 0.535i)3-s − 6.73i·4-s + 2.48·6-s + (2.25 − 2.25i)7-s + (−6.32 − 6.32i)8-s − 8.42i·9-s + 3.31·11-s + (3.60 − 3.60i)12-s + (−7.80 − 7.80i)13-s − 10.4i·14-s − 2.37·16-s + (6.60 − 6.60i)17-s + (−19.5 − 19.5i)18-s + 14.8i·19-s + ⋯
L(s)  = 1  + (1.15 − 1.15i)2-s + (0.178 + 0.178i)3-s − 1.68i·4-s + 0.413·6-s + (0.322 − 0.322i)7-s + (−0.790 − 0.790i)8-s − 0.936i·9-s + 0.301·11-s + (0.300 − 0.300i)12-s + (−0.600 − 0.600i)13-s − 0.746i·14-s − 0.148·16-s + (0.388 − 0.388i)17-s + (−1.08 − 1.08i)18-s + 0.780i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ -0.437 + 0.899i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.65891 - 2.65219i\)
\(L(\frac12)\) \(\approx\) \(1.65891 - 2.65219i\)
\(L(2)\) 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 \( 1 - 3.31T \)
good2 \( 1 + (-2.31 + 2.31i)T - 4iT^{2} \)
3 \( 1 + (-0.535 - 0.535i)T + 9iT^{2} \)
7 \( 1 + (-2.25 + 2.25i)T - 49iT^{2} \)
13 \( 1 + (7.80 + 7.80i)T + 169iT^{2} \)
17 \( 1 + (-6.60 + 6.60i)T - 289iT^{2} \)
19 \( 1 - 14.8iT - 361T^{2} \)
23 \( 1 + (6.20 + 6.20i)T + 529iT^{2} \)
29 \( 1 - 55.3iT - 841T^{2} \)
31 \( 1 - 7.07T + 961T^{2} \)
37 \( 1 + (22.2 - 22.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 36.8T + 1.68e3T^{2} \)
43 \( 1 + (-50.8 - 50.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (-45.9 + 45.9i)T - 2.20e3iT^{2} \)
53 \( 1 + (5.18 + 5.18i)T + 2.80e3iT^{2} \)
59 \( 1 - 94.6iT - 3.48e3T^{2} \)
61 \( 1 - 102.T + 3.72e3T^{2} \)
67 \( 1 + (16.0 - 16.0i)T - 4.48e3iT^{2} \)
71 \( 1 + 12.0T + 5.04e3T^{2} \)
73 \( 1 + (86.2 + 86.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 76.3iT - 6.24e3T^{2} \)
83 \( 1 + (85.3 + 85.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 3.80iT - 7.92e3T^{2} \)
97 \( 1 + (-33.8 + 33.8i)T - 9.40e3iT^{2} \)
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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

−11.66688815990990489418617480598, −10.51781241011726802081063623087, −9.944363188392197417294816500698, −8.714045727225361229623540656247, −7.30074321035356445883425817192, −5.92060485924030518909459997742, −4.85763826825393482000281793903, −3.80207140956581736346715739327, −2.88874390002670120697812287687, −1.23839852668315984237716646297, 2.30947563563979532927108027682, 3.98039752538977825672583443558, 4.95645395908749390891273453487, 5.85193303242710105467622673409, 6.99520755467412625276510131088, 7.74805057898046350953195211542, 8.679057569176658835634994263814, 10.03610334443684071200503220992, 11.39435362804233723750096814446, 12.25717556506491997559469307462

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