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Properties

Label 2-320-5.4-c5-0-16
Degree $2$
Conductor $320$
Sign $-0.303 - 0.952i$
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  + 9.81i·3-s + (16.9 + 53.2i)5-s − 222. i·7-s + 146.·9-s − 407.·11-s + 465. i·13-s + (−522. + 166. i)15-s + 284. i·17-s + 323.·19-s + 2.18e3·21-s + 12.0i·23-s + (−2.54e3 + 1.80e3i)25-s + 3.82e3i·27-s + 4.38e3·29-s + 1.01e4·31-s + ⋯
L(s)  = 1  + 0.629i·3-s + (0.303 + 0.952i)5-s − 1.71i·7-s + 0.603·9-s − 1.01·11-s + 0.764i·13-s + (−0.600 + 0.191i)15-s + 0.238i·17-s + 0.205·19-s + 1.08·21-s + 0.00475i·23-s + (−0.815 + 0.578i)25-s + 1.00i·27-s + 0.967·29-s + 1.89·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.303 - 0.952i$
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: \((2,\ 320,\ (\ :5/2),\ -0.303 - 0.952i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.817055559\)
\(L(\frac12)\) \(\approx\) \(1.817055559\)
\(L(\frac{7}{2})\) 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 \( 1 + (-16.9 - 53.2i)T \)
good3 \( 1 - 9.81iT - 243T^{2} \)
7 \( 1 + 222. iT - 1.68e4T^{2} \)
11 \( 1 + 407.T + 1.61e5T^{2} \)
13 \( 1 - 465. iT - 3.71e5T^{2} \)
17 \( 1 - 284. iT - 1.41e6T^{2} \)
19 \( 1 - 323.T + 2.47e6T^{2} \)
23 \( 1 - 12.0iT - 6.43e6T^{2} \)
29 \( 1 - 4.38e3T + 2.05e7T^{2} \)
31 \( 1 - 1.01e4T + 2.86e7T^{2} \)
37 \( 1 - 6.90e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.07e4T + 1.15e8T^{2} \)
43 \( 1 + 6.00e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.50e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.05e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.09e4T + 7.14e8T^{2} \)
61 \( 1 - 3.43e4T + 8.44e8T^{2} \)
67 \( 1 - 4.58e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.80e4T + 1.80e9T^{2} \)
73 \( 1 - 6.86e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.50e4T + 3.07e9T^{2} \)
83 \( 1 + 5.10e4iT - 3.93e9T^{2} \)
89 \( 1 - 5.94e4T + 5.58e9T^{2} \)
97 \( 1 - 7.31e4iT - 8.58e9T^{2} \)
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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

−10.61361855455661962619050846908, −10.34814296040525848110565994392, −9.643195087507939518033196679426, −8.057508510206213147969224659584, −7.15561149110447118672685073904, −6.44246589432021491282849158999, −4.81590135611348469364598833674, −4.01637437719569591190788212117, −2.86936577240353826526540091697, −1.22263995173375991640894083901, 0.50659367851999796886382718292, 1.86865999638875692263635379943, 2.83104221609183293058594456375, 4.79503769059346766936796752513, 5.50120809825822444558669388719, 6.48274154157432662850574017200, 7.947336103755774100878451869964, 8.448146192249296060981237862547, 9.519997012443574042743926085663, 10.36130422593573606886707617209

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