Properties

Label 2-320-1.1-c3-0-12
Degree $2$
Conductor $320$
Sign $1$
Analytic cond. $18.8806$
Root an. cond. $4.34518$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.47·3-s + 5·5-s + 31.3·7-s − 6.99·9-s − 8.94·11-s + 62·13-s + 22.3·15-s − 46·17-s + 107.·19-s + 140·21-s − 192.·23-s + 25·25-s − 152.·27-s + 90·29-s + 152.·31-s − 40.0·33-s + 156.·35-s + 214·37-s + 277.·39-s − 10·41-s − 67.0·43-s − 34.9·45-s − 398.·47-s + 637.·49-s − 205.·51-s + 678·53-s − 44.7·55-s + ⋯
L(s)  = 1  + 0.860·3-s + 0.447·5-s + 1.69·7-s − 0.259·9-s − 0.245·11-s + 1.32·13-s + 0.384·15-s − 0.656·17-s + 1.29·19-s + 1.45·21-s − 1.74·23-s + 0.200·25-s − 1.08·27-s + 0.576·29-s + 0.880·31-s − 0.211·33-s + 0.755·35-s + 0.950·37-s + 1.13·39-s − 0.0380·41-s − 0.237·43-s − 0.115·45-s − 1.23·47-s + 1.85·49-s − 0.564·51-s + 1.75·53-s − 0.109·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $1$
Analytic conductor: \(18.8806\)
Root analytic conductor: \(4.34518\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.186747943\)
\(L(\frac12)\) \(\approx\) \(3.186747943\)
\(L(\frac{5}{2})\) 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 \( 1 - 5T \)
good3 \( 1 - 4.47T + 27T^{2} \)
7 \( 1 - 31.3T + 343T^{2} \)
11 \( 1 + 8.94T + 1.33e3T^{2} \)
13 \( 1 - 62T + 2.19e3T^{2} \)
17 \( 1 + 46T + 4.91e3T^{2} \)
19 \( 1 - 107.T + 6.85e3T^{2} \)
23 \( 1 + 192.T + 1.21e4T^{2} \)
29 \( 1 - 90T + 2.43e4T^{2} \)
31 \( 1 - 152.T + 2.97e4T^{2} \)
37 \( 1 - 214T + 5.06e4T^{2} \)
41 \( 1 + 10T + 6.89e4T^{2} \)
43 \( 1 + 67.0T + 7.95e4T^{2} \)
47 \( 1 + 398.T + 1.03e5T^{2} \)
53 \( 1 - 678T + 1.48e5T^{2} \)
59 \( 1 + 411.T + 2.05e5T^{2} \)
61 \( 1 + 250T + 2.26e5T^{2} \)
67 \( 1 - 49.1T + 3.00e5T^{2} \)
71 \( 1 - 366.T + 3.57e5T^{2} \)
73 \( 1 - 522T + 3.89e5T^{2} \)
79 \( 1 + 876.T + 4.93e5T^{2} \)
83 \( 1 - 380.T + 5.71e5T^{2} \)
89 \( 1 - 970T + 7.04e5T^{2} \)
97 \( 1 + 934T + 9.12e5T^{2} \)
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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

−11.28904346255545418682715934292, −10.26262158657310536041491353365, −9.121500481511738986138752906983, −8.248677103741283532669548678471, −7.83709211555687191639217150466, −6.22585159214282625388433696280, −5.16826921682130328249364668209, −3.95065408755848868917436086590, −2.52041268001832373920895730493, −1.39120333416588061945260567780, 1.39120333416588061945260567780, 2.52041268001832373920895730493, 3.95065408755848868917436086590, 5.16826921682130328249364668209, 6.22585159214282625388433696280, 7.83709211555687191639217150466, 8.248677103741283532669548678471, 9.121500481511738986138752906983, 10.26262158657310536041491353365, 11.28904346255545418682715934292

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