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Properties

Label 2-320-40.29-c5-0-15
Degree $2$
Conductor $320$
Sign $0.897 - 0.441i$
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  − 10.4·3-s + (17.9 − 52.9i)5-s − 86.7i·7-s − 134.·9-s + 258. i·11-s − 465.·13-s + (−187. + 551. i)15-s + 2.02e3i·17-s + 1.05e3i·19-s + 904. i·21-s − 3.99e3i·23-s + (−2.47e3 − 1.90e3i)25-s + 3.93e3·27-s + 3.24e3i·29-s + 991.·31-s + ⋯
L(s)  = 1  − 0.668·3-s + (0.321 − 0.946i)5-s − 0.669i·7-s − 0.552·9-s + 0.644i·11-s − 0.763·13-s + (−0.215 + 0.633i)15-s + 1.70i·17-s + 0.667i·19-s + 0.447i·21-s − 1.57i·23-s + (−0.792 − 0.609i)25-s + 1.03·27-s + 0.715i·29-s + 0.185·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.897 - 0.441i$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ 0.897 - 0.441i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.078685674\)
\(L(\frac12)\) \(\approx\) \(1.078685674\)
\(L(\frac{7}{2})\) 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 \( 1 + (-17.9 + 52.9i)T \)
good3 \( 1 + 10.4T + 243T^{2} \)
7 \( 1 + 86.7iT - 1.68e4T^{2} \)
11 \( 1 - 258. iT - 1.61e5T^{2} \)
13 \( 1 + 465.T + 3.71e5T^{2} \)
17 \( 1 - 2.02e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.05e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.99e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.24e3iT - 2.05e7T^{2} \)
31 \( 1 - 991.T + 2.86e7T^{2} \)
37 \( 1 + 8.82e3T + 6.93e7T^{2} \)
41 \( 1 - 5.63e3T + 1.15e8T^{2} \)
43 \( 1 + 1.77e3T + 1.47e8T^{2} \)
47 \( 1 + 1.77e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.59e4T + 4.18e8T^{2} \)
59 \( 1 - 1.81e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.54e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.20e4T + 1.35e9T^{2} \)
71 \( 1 - 7.13e4T + 1.80e9T^{2} \)
73 \( 1 + 3.25e3iT - 2.07e9T^{2} \)
79 \( 1 - 8.42e4T + 3.07e9T^{2} \)
83 \( 1 - 7.90e4T + 3.93e9T^{2} \)
89 \( 1 + 6.51e4T + 5.58e9T^{2} \)
97 \( 1 - 3.39e4iT - 8.58e9T^{2} \)
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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

−10.58920752544919259282429439460, −10.25497091415937063694865953839, −8.911463231436897258450703324796, −8.140617402241796179295660337162, −6.86652248847377915975185362893, −5.85493990591381929390846458197, −4.93018983092607970733997262543, −3.96504698178832432732878747374, −2.11130631221896629197328060309, −0.800509889831658151805998370560, 0.42428013887484635134782145604, 2.36975037305551694391749799927, 3.21234866790597110239998604371, 5.06451554077858606090821763870, 5.74418933754325718542566469953, 6.74627355106662845745434039559, 7.68102430295231265530186842119, 9.069949266918068779212676179872, 9.785571951681569255187872897192, 11.01021394970209124115176744703

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