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Properties

Label 2-55-55.32-c5-0-8
Degree $2$
Conductor $55$
Sign $-0.999 + 0.0185i$
Analytic cond. $8.82111$
Root an. cond. $2.97003$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.8 + 13.8i)3-s + 32i·4-s + (48.0 + 28.5i)5-s − 140. i·9-s − 401.·11-s + (−442. − 442. i)12-s + (−1.06e3 + 271. i)15-s − 1.02e3·16-s + (−912 + 1.53e3i)20-s + (1.99e3 − 1.99e3i)23-s + (1.50e3 + 2.74e3i)25-s + (−1.42e3 − 1.42e3i)27-s − 7.35e3·31-s + (5.55e3 − 5.55e3i)33-s + 4.48e3·36-s + (7.66e3 + 7.66e3i)37-s + ⋯
L(s)  = 1  + (−0.887 + 0.887i)3-s + i·4-s + (0.860 + 0.509i)5-s − 0.576i·9-s − 1.00·11-s + (−0.887 − 0.887i)12-s + (−1.21 + 0.311i)15-s − 16-s + (−0.509 + 0.860i)20-s + (0.787 − 0.787i)23-s + (0.480 + 0.877i)25-s + (−0.375 − 0.375i)27-s − 1.37·31-s + (0.887 − 0.887i)33-s + 0.576·36-s + (0.921 + 0.921i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0185i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.999 + 0.0185i$
Analytic conductor: \(8.82111\)
Root analytic conductor: \(2.97003\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :5/2),\ -0.999 + 0.0185i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.00892556 - 0.959981i\)
\(L(\frac12)\) \(\approx\) \(0.00892556 - 0.959981i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-48.0 - 28.5i)T \)
11 \( 1 + 401.T \)
good2 \( 1 - 32iT^{2} \)
3 \( 1 + (13.8 - 13.8i)T - 243iT^{2} \)
7 \( 1 - 1.68e4iT^{2} \)
13 \( 1 + 3.71e5iT^{2} \)
17 \( 1 - 1.41e6iT^{2} \)
19 \( 1 + 2.47e6T^{2} \)
23 \( 1 + (-1.99e3 + 1.99e3i)T - 6.43e6iT^{2} \)
29 \( 1 + 2.05e7T^{2} \)
31 \( 1 + 7.35e3T + 2.86e7T^{2} \)
37 \( 1 + (-7.66e3 - 7.66e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.15e8T^{2} \)
43 \( 1 + 1.47e8iT^{2} \)
47 \( 1 + (-2.11e4 - 2.11e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (6.66e3 - 6.66e3i)T - 4.18e8iT^{2} \)
59 \( 1 - 4.73e4iT - 7.14e8T^{2} \)
61 \( 1 - 8.44e8T^{2} \)
67 \( 1 + (3.18e4 + 3.18e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 6.62e4T + 1.80e9T^{2} \)
73 \( 1 + 2.07e9iT^{2} \)
79 \( 1 + 3.07e9T^{2} \)
83 \( 1 + 3.93e9iT^{2} \)
89 \( 1 - 9.10e4iT - 5.58e9T^{2} \)
97 \( 1 + (-3.76e4 - 3.76e4i)T + 8.58e9iT^{2} \)
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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

−15.07599136520066529446306432804, −13.59041376854581935311465881156, −12.58550078953130478324592457830, −11.16007211575933167700853925739, −10.44809324729893950189435982432, −9.145166513384579021897552131720, −7.47958120468085375775936831234, −5.92659161993122784020127345261, −4.59785915703742243021769629018, −2.78832256560050963407393932376, 0.51353528424857197713019073309, 1.88084474258109118464182466984, 5.21036468498612630645065562287, 5.89011433969331129648823448050, 7.19935782064137758785701602019, 9.106849533811250561300834339886, 10.32718121200992618957336514724, 11.38713021230586649525019152477, 12.80001397938308977001417922736, 13.44450160843466266420681290553

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