Properties

Label 2-320-1.1-c5-0-13
Degree $2$
Conductor $320$
Sign $1$
Analytic cond. $51.3228$
Root an. cond. $7.16399$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22·3-s + 25·5-s + 218·7-s + 241·9-s + 480·11-s + 622·13-s − 550·15-s + 186·17-s + 1.20e3·19-s − 4.79e3·21-s − 3.18e3·23-s + 625·25-s + 44·27-s − 5.52e3·29-s + 9.35e3·31-s − 1.05e4·33-s + 5.45e3·35-s − 5.61e3·37-s − 1.36e4·39-s − 1.43e4·41-s + 370·43-s + 6.02e3·45-s + 1.61e4·47-s + 3.07e4·49-s − 4.09e3·51-s + 4.37e3·53-s + 1.20e4·55-s + ⋯
L(s)  = 1  − 1.41·3-s + 0.447·5-s + 1.68·7-s + 0.991·9-s + 1.19·11-s + 1.02·13-s − 0.631·15-s + 0.156·17-s + 0.765·19-s − 2.37·21-s − 1.25·23-s + 1/5·25-s + 0.0116·27-s − 1.22·29-s + 1.74·31-s − 1.68·33-s + 0.752·35-s − 0.674·37-s − 1.44·39-s − 1.33·41-s + 0.0305·43-s + 0.443·45-s + 1.06·47-s + 1.82·49-s − 0.220·51-s + 0.213·53-s + 0.534·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $1$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.000045542\)
\(L(\frac12)\) \(\approx\) \(2.000045542\)
\(L(\frac{7}{2})\) 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 \( 1 - p^{2} T \)
good3 \( 1 + 22 T + p^{5} T^{2} \)
7 \( 1 - 218 T + p^{5} T^{2} \)
11 \( 1 - 480 T + p^{5} T^{2} \)
13 \( 1 - 622 T + p^{5} T^{2} \)
17 \( 1 - 186 T + p^{5} T^{2} \)
19 \( 1 - 1204 T + p^{5} T^{2} \)
23 \( 1 + 3186 T + p^{5} T^{2} \)
29 \( 1 + 5526 T + p^{5} T^{2} \)
31 \( 1 - 9356 T + p^{5} T^{2} \)
37 \( 1 + 5618 T + p^{5} T^{2} \)
41 \( 1 + 14394 T + p^{5} T^{2} \)
43 \( 1 - 370 T + p^{5} T^{2} \)
47 \( 1 - 16146 T + p^{5} T^{2} \)
53 \( 1 - 4374 T + p^{5} T^{2} \)
59 \( 1 - 11748 T + p^{5} T^{2} \)
61 \( 1 + 13202 T + p^{5} T^{2} \)
67 \( 1 - 11542 T + p^{5} T^{2} \)
71 \( 1 + 29532 T + p^{5} T^{2} \)
73 \( 1 - 33698 T + p^{5} T^{2} \)
79 \( 1 - 31208 T + p^{5} T^{2} \)
83 \( 1 - 38466 T + p^{5} T^{2} \)
89 \( 1 - 119514 T + p^{5} T^{2} \)
97 \( 1 - 94658 T + p^{5} T^{2} \)
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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

−10.98668541700642553582210617146, −10.17185805539943443464370039470, −8.910625215499231818669415826326, −7.911277503459618332483245527739, −6.64260677278629920410481616865, −5.80351499906646376029988529386, −5.00607540471512156675270326931, −3.94334560162021756372403947958, −1.73695807122594335718778322397, −0.938276032071123315674735698199, 0.938276032071123315674735698199, 1.73695807122594335718778322397, 3.94334560162021756372403947958, 5.00607540471512156675270326931, 5.80351499906646376029988529386, 6.64260677278629920410481616865, 7.911277503459618332483245527739, 8.910625215499231818669415826326, 10.17185805539943443464370039470, 10.98668541700642553582210617146

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