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Properties

Label 2-320-20.19-c8-0-87
Degree $2$
Conductor $320$
Sign $-0.939 - 0.343i$
Analytic cond. $130.361$
Root an. cond. $11.4175$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 77.4·3-s + (−586. − 214. i)5-s + 2.76e3·7-s − 560.·9-s − 1.87e4i·11-s + 3.86e4i·13-s + (−4.54e4 − 1.66e4i)15-s − 7.64e4i·17-s − 3.35e4i·19-s + 2.14e5·21-s − 2.22e5·23-s + (2.98e5 + 2.52e5i)25-s − 5.51e5·27-s − 6.51e5·29-s + 1.03e6i·31-s + ⋯
L(s)  = 1  + 0.956·3-s + (−0.939 − 0.343i)5-s + 1.15·7-s − 0.0853·9-s − 1.28i·11-s + 1.35i·13-s + (−0.898 − 0.328i)15-s − 0.915i·17-s − 0.257i·19-s + 1.10·21-s − 0.794·23-s + (0.763 + 0.645i)25-s − 1.03·27-s − 0.921·29-s + 1.12i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.939 - 0.343i$
Analytic conductor: \(130.361\)
Root analytic conductor: \(11.4175\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :4),\ -0.939 - 0.343i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.04956284235\)
\(L(\frac12)\) \(\approx\) \(0.04956284235\)
\(L(5)\) 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 \( 1 + (586. + 214. i)T \)
good3 \( 1 - 77.4T + 6.56e3T^{2} \)
7 \( 1 - 2.76e3T + 5.76e6T^{2} \)
11 \( 1 + 1.87e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.86e4iT - 8.15e8T^{2} \)
17 \( 1 + 7.64e4iT - 6.97e9T^{2} \)
19 \( 1 + 3.35e4iT - 1.69e10T^{2} \)
23 \( 1 + 2.22e5T + 7.83e10T^{2} \)
29 \( 1 + 6.51e5T + 5.00e11T^{2} \)
31 \( 1 - 1.03e6iT - 8.52e11T^{2} \)
37 \( 1 + 2.60e6iT - 3.51e12T^{2} \)
41 \( 1 - 4.22e6T + 7.98e12T^{2} \)
43 \( 1 + 8.65e5T + 1.16e13T^{2} \)
47 \( 1 - 1.55e6T + 2.38e13T^{2} \)
53 \( 1 - 1.13e7iT - 6.22e13T^{2} \)
59 \( 1 + 1.73e7iT - 1.46e14T^{2} \)
61 \( 1 + 6.27e6T + 1.91e14T^{2} \)
67 \( 1 + 3.69e7T + 4.06e14T^{2} \)
71 \( 1 - 2.42e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.21e7iT - 8.06e14T^{2} \)
79 \( 1 + 3.05e7iT - 1.51e15T^{2} \)
83 \( 1 + 2.71e7T + 2.25e15T^{2} \)
89 \( 1 + 3.34e7T + 3.93e15T^{2} \)
97 \( 1 - 1.14e8iT - 7.83e15T^{2} \)
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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

−9.235724353263448385124108838329, −8.777640927117356306575594708329, −7.966115194172696354197433803988, −7.23882435039094041472440587316, −5.68576298296463952385000425954, −4.50638832524814602158030871810, −3.66259981130510599385486824064, −2.52060350576409662478006331878, −1.30855514593162947278026540340, −0.008232220259879718991774405398, 1.59971128762966367007996832406, 2.61323570064546375865286554816, 3.73211375413447736558395365679, 4.59522252003715032764678727527, 5.89631632007369846982542311418, 7.54183270513011193125040138510, 7.85164729144997182785827785813, 8.581794250773126649636940446718, 9.844198604120799528866496911270, 10.75110189847664418186770027467

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