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Properties

Label 2-320-5.4-c7-0-54
Degree $2$
Conductor $320$
Sign $-0.850 + 0.526i$
Analytic cond. $99.9632$
Root an. cond. $9.99816$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 83.6i·3-s + (−237. + 147. i)5-s + 185. i·7-s − 4.81e3·9-s + 3.56e3·11-s − 6.09e3i·13-s + (1.23e4 + 1.98e4i)15-s + 1.24e4i·17-s + 5.06e4·19-s + 1.55e4·21-s − 1.14e4i·23-s + (3.48e4 − 6.99e4i)25-s + 2.19e5i·27-s + 1.01e5·29-s + 2.90e4·31-s + ⋯
L(s)  = 1  − 1.78i·3-s + (−0.850 + 0.526i)5-s + 0.204i·7-s − 2.20·9-s + 0.806·11-s − 0.769i·13-s + (0.941 + 1.52i)15-s + 0.615i·17-s + 1.69·19-s + 0.366·21-s − 0.196i·23-s + (0.446 − 0.894i)25-s + 2.15i·27-s + 0.776·29-s + 0.175·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.850 + 0.526i$
Analytic conductor: \(99.9632\)
Root analytic conductor: \(9.99816\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :7/2),\ -0.850 + 0.526i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.574198792\)
\(L(\frac12)\) \(\approx\) \(1.574198792\)
\(L(\frac{9}{2})\) 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 \( 1 + (237. - 147. i)T \)
good3 \( 1 + 83.6iT - 2.18e3T^{2} \)
7 \( 1 - 185. iT - 8.23e5T^{2} \)
11 \( 1 - 3.56e3T + 1.94e7T^{2} \)
13 \( 1 + 6.09e3iT - 6.27e7T^{2} \)
17 \( 1 - 1.24e4iT - 4.10e8T^{2} \)
19 \( 1 - 5.06e4T + 8.93e8T^{2} \)
23 \( 1 + 1.14e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.01e5T + 1.72e10T^{2} \)
31 \( 1 - 2.90e4T + 2.75e10T^{2} \)
37 \( 1 - 1.49e5iT - 9.49e10T^{2} \)
41 \( 1 + 3.74e5T + 1.94e11T^{2} \)
43 \( 1 + 1.74e5iT - 2.71e11T^{2} \)
47 \( 1 + 4.28e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.71e6iT - 1.17e12T^{2} \)
59 \( 1 + 1.34e5T + 2.48e12T^{2} \)
61 \( 1 - 1.39e6T + 3.14e12T^{2} \)
67 \( 1 - 2.60e6iT - 6.06e12T^{2} \)
71 \( 1 - 4.91e6T + 9.09e12T^{2} \)
73 \( 1 + 1.19e5iT - 1.10e13T^{2} \)
79 \( 1 + 4.70e6T + 1.92e13T^{2} \)
83 \( 1 + 9.19e6iT - 2.71e13T^{2} \)
89 \( 1 + 6.43e6T + 4.42e13T^{2} \)
97 \( 1 - 1.26e7iT - 8.07e13T^{2} \)
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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

−10.15185119690414391748641416888, −8.632815334396434708004016426920, −7.991645425642458566023729195321, −7.12525621542595213397748765238, −6.49770087695684684415729710623, −5.38107157662990109691057549271, −3.58891093678567826642657072721, −2.63805890681710981305660512848, −1.32714013589441710935654610596, −0.44832960243000669625195613030, 0.961845713744679897653751052412, 3.05680423702925978953488880509, 3.94354485115344401050257979748, 4.63026008000880946116307414342, 5.51415976898761875404783157469, 7.03994711080581731908456989445, 8.267458308211127405825922937840, 9.264192023460179320597409171519, 9.617320301755766315702429140264, 10.83364992395010074077887949450

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