Properties

Degree $2$
Conductor $320$
Sign $0.106 + 0.994i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 19.1·3-s + (55.2 + 8.66i)5-s − 121. i·7-s + 123.·9-s − 298i·11-s − 485.·13-s + (1.05e3 + 165. i)15-s − 420. i·17-s − 2.57e3i·19-s − 2.32e3i·21-s − 717. i·23-s + (2.97e3 + 956. i)25-s − 2.29e3·27-s − 3.73e3i·29-s − 6.22e3·31-s + ⋯
L(s)  = 1  + 1.22·3-s + (0.987 + 0.154i)5-s − 0.937i·7-s + 0.506·9-s − 0.742i·11-s − 0.797·13-s + (1.21 + 0.190i)15-s − 0.353i·17-s − 1.63i·19-s − 1.15i·21-s − 0.282i·23-s + (0.952 + 0.306i)25-s − 0.606·27-s − 0.824i·29-s − 1.16·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.106 + 0.994i$
Motivic weight: \(5\)
Character: $\chi_{320} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ 0.106 + 0.994i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.316659840\)
\(L(\frac12)\) \(\approx\) \(3.316659840\)
\(L(\frac{7}{2})\) 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 \( 1 + (-55.2 - 8.66i)T \)
good3 \( 1 - 19.1T + 243T^{2} \)
7 \( 1 + 121. iT - 1.68e4T^{2} \)
11 \( 1 + 298iT - 1.61e5T^{2} \)
13 \( 1 + 485.T + 3.71e5T^{2} \)
17 \( 1 + 420. iT - 1.41e6T^{2} \)
19 \( 1 + 2.57e3iT - 2.47e6T^{2} \)
23 \( 1 + 717. iT - 6.43e6T^{2} \)
29 \( 1 + 3.73e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.22e3T + 2.86e7T^{2} \)
37 \( 1 + 5.72e3T + 6.93e7T^{2} \)
41 \( 1 - 1.13e4T + 1.15e8T^{2} \)
43 \( 1 - 1.91e4T + 1.47e8T^{2} \)
47 \( 1 - 2.28e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.72e4T + 4.18e8T^{2} \)
59 \( 1 - 1.63e4iT - 7.14e8T^{2} \)
61 \( 1 - 4.87e4iT - 8.44e8T^{2} \)
67 \( 1 - 8.39e3T + 1.35e9T^{2} \)
71 \( 1 - 2.90e4T + 1.80e9T^{2} \)
73 \( 1 + 420. iT - 2.07e9T^{2} \)
79 \( 1 - 3.11e4T + 3.07e9T^{2} \)
83 \( 1 - 1.06e5T + 3.93e9T^{2} \)
89 \( 1 + 9.93e4T + 5.58e9T^{2} \)
97 \( 1 + 1.80e5iT - 8.58e9T^{2} \)
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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

−10.45965502202216185676783158561, −9.395797792137018107239071156176, −8.953434145220363689328340917208, −7.68639869443675031890466075771, −6.94035889141567355180883906464, −5.63656289124307296610401937084, −4.32614870012673123857644889375, −3.01759783965403678094677442767, −2.24840760247956606389440022708, −0.67476676310681496254354079087, 1.76831791927789783016103672179, 2.36706606272904450104723339505, 3.60573693998182465514884394674, 5.12664824199661138020422413285, 6.02931760482895390786156342259, 7.39628913609806333830158456538, 8.323146710167734783241895210654, 9.292112446204157633397863598654, 9.651584132717026155778158852722, 10.79452282952754876611579202862

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