Properties

Label 2-320-5.4-c5-0-51
Degree $2$
Conductor $320$
Sign $-0.983 - 0.178i$
Analytic cond. $51.3228$
Root an. cond. $7.16399$
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  − 14i·3-s + (−55 − 10i)5-s − 158i·7-s + 47·9-s + 148·11-s − 684i·13-s + (−140 + 770i)15-s − 2.04e3i·17-s + 2.22e3·19-s − 2.21e3·21-s − 1.24e3i·23-s + (2.92e3 + 1.10e3i)25-s − 4.06e3i·27-s − 270·29-s − 2.04e3·31-s + ⋯
L(s)  = 1  − 0.898i·3-s + (−0.983 − 0.178i)5-s − 1.21i·7-s + 0.193·9-s + 0.368·11-s − 1.12i·13-s + (−0.160 + 0.883i)15-s − 1.71i·17-s + 1.41·19-s − 1.09·21-s − 0.491i·23-s + (0.936 + 0.352i)25-s − 1.07i·27-s − 0.0596·29-s − 0.382·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.983 - 0.178i$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ -0.983 - 0.178i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.547947274\)
\(L(\frac12)\) \(\approx\) \(1.547947274\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (55 + 10i)T \)
good3 \( 1 + 14iT - 243T^{2} \)
7 \( 1 + 158iT - 1.68e4T^{2} \)
11 \( 1 - 148T + 1.61e5T^{2} \)
13 \( 1 + 684iT - 3.71e5T^{2} \)
17 \( 1 + 2.04e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.22e3T + 2.47e6T^{2} \)
23 \( 1 + 1.24e3iT - 6.43e6T^{2} \)
29 \( 1 + 270T + 2.05e7T^{2} \)
31 \( 1 + 2.04e3T + 2.86e7T^{2} \)
37 \( 1 + 4.37e3iT - 6.93e7T^{2} \)
41 \( 1 + 2.39e3T + 1.15e8T^{2} \)
43 \( 1 + 2.29e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.06e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.96e3iT - 4.18e8T^{2} \)
59 \( 1 + 3.97e4T + 7.14e8T^{2} \)
61 \( 1 - 4.22e4T + 8.44e8T^{2} \)
67 \( 1 - 3.20e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.24e3T + 1.80e9T^{2} \)
73 \( 1 - 3.01e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.52e4T + 3.07e9T^{2} \)
83 \( 1 - 2.78e4iT - 3.93e9T^{2} \)
89 \( 1 - 8.52e4T + 5.58e9T^{2} \)
97 \( 1 - 9.72e4iT - 8.58e9T^{2} \)
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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

−10.40798133918283463611814069135, −9.362097390708126835512903973996, −7.976905079686492485291398375542, −7.41206681584715330809595035596, −6.85627125866067399252101557287, −5.22660877959376333887606380084, −4.09000221554575593486135417789, −2.96737626266581221367272840611, −1.06214537871068794523823957956, −0.50331916655101393402041627034, 1.64816353048125030239897637389, 3.31417070250116321552865347618, 4.11918762105717064503094414208, 5.18043803815000671230890681206, 6.39042117936015941783695381724, 7.56174395117571142419891896144, 8.684629844601267694369339341083, 9.341510265879589143710613731559, 10.34757264904187364532269775074, 11.40152723005750657822842957554

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