Properties

Label 2-320-1.1-c1-0-0
Degree $2$
Conductor $320$
Sign $1$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·3-s − 5-s − 2.82·7-s + 5.00·9-s + 5.65·11-s + 2·13-s + 2.82·15-s + 2·17-s + 8.00·21-s + 2.82·23-s + 25-s − 5.65·27-s − 6·29-s + 5.65·31-s − 16.0·33-s + 2.82·35-s + 10·37-s − 5.65·39-s + 2·41-s + 8.48·43-s − 5.00·45-s − 2.82·47-s + 1.00·49-s − 5.65·51-s − 6·53-s − 5.65·55-s − 11.3·59-s + ⋯
L(s)  = 1  − 1.63·3-s − 0.447·5-s − 1.06·7-s + 1.66·9-s + 1.70·11-s + 0.554·13-s + 0.730·15-s + 0.485·17-s + 1.74·21-s + 0.589·23-s + 0.200·25-s − 1.08·27-s − 1.11·29-s + 1.01·31-s − 2.78·33-s + 0.478·35-s + 1.64·37-s − 0.905·39-s + 0.312·41-s + 1.29·43-s − 0.745·45-s − 0.412·47-s + 0.142·49-s − 0.792·51-s − 0.824·53-s − 0.762·55-s − 1.47·59-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $1$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6869533875\)
\(L(\frac12)\) \(\approx\) \(0.6869533875\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 + 2.82T + 3T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 2.82T + 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−11.54096329063596514712291905022, −11.03111970956455570875612605421, −9.862515474013922215550479093911, −9.096978939488460814703141204352, −7.48765597332836045376350376470, −6.38912401683433574009480666981, −6.06943653388910276154682339658, −4.59789173031085180214409789496, −3.54943286150140438787827956806, −0.942019062778363645446364015230, 0.942019062778363645446364015230, 3.54943286150140438787827956806, 4.59789173031085180214409789496, 6.06943653388910276154682339658, 6.38912401683433574009480666981, 7.48765597332836045376350376470, 9.096978939488460814703141204352, 9.862515474013922215550479093911, 11.03111970956455570875612605421, 11.54096329063596514712291905022

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