Properties

Label 2-320-20.19-c2-0-14
Degree $2$
Conductor $320$
Sign $1$
Analytic cond. $8.71936$
Root an. cond. $2.95285$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 5·5-s + 4·7-s + 7·9-s + 20·15-s + 16·21-s − 44·23-s + 25·25-s − 8·27-s + 22·29-s + 20·35-s + 62·41-s − 76·43-s + 35·45-s + 4·47-s − 33·49-s + 58·61-s + 28·63-s + 116·67-s − 176·69-s + 100·75-s − 95·81-s − 76·83-s + 88·87-s − 142·89-s − 122·101-s − 44·103-s + ⋯
L(s)  = 1  + 4/3·3-s + 5-s + 4/7·7-s + 7/9·9-s + 4/3·15-s + 0.761·21-s − 1.91·23-s + 25-s − 0.296·27-s + 0.758·29-s + 4/7·35-s + 1.51·41-s − 1.76·43-s + 7/9·45-s + 4/47·47-s − 0.673·49-s + 0.950·61-s + 4/9·63-s + 1.73·67-s − 2.55·69-s + 4/3·75-s − 1.17·81-s − 0.915·83-s + 1.01·87-s − 1.59·89-s − 1.20·101-s − 0.427·103-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $1$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{320} (319, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.015301371\)
\(L(\frac12)\) \(\approx\) \(3.015301371\)
\(L(2)\) 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 \( 1 - p T \)
good3 \( 1 - 4 T + p^{2} T^{2} \)
7 \( 1 - 4 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( ( 1 - p T )( 1 + p T ) \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( ( 1 - p T )( 1 + p T ) \)
23 \( 1 + 44 T + p^{2} T^{2} \)
29 \( 1 - 22 T + p^{2} T^{2} \)
31 \( ( 1 - p T )( 1 + p T ) \)
37 \( ( 1 - p T )( 1 + p T ) \)
41 \( 1 - 62 T + p^{2} T^{2} \)
43 \( 1 + 76 T + p^{2} T^{2} \)
47 \( 1 - 4 T + p^{2} T^{2} \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 - 58 T + p^{2} T^{2} \)
67 \( 1 - 116 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( ( 1 - p T )( 1 + p T ) \)
79 \( ( 1 - p T )( 1 + p T ) \)
83 \( 1 + 76 T + p^{2} T^{2} \)
89 \( 1 + 142 T + p^{2} T^{2} \)
97 \( ( 1 - p T )( 1 + p T ) \)
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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

−11.36557375328696613402873798062, −10.14242724505272250474643863845, −9.551201446942913015030291934024, −8.509081267817681463388950779282, −7.934971227553052378192820105367, −6.61069813333769198944009927879, −5.43604378190078653650128548413, −4.08047292942168149987346636493, −2.70961430055421932406734153548, −1.73738228925401918617574601968, 1.73738228925401918617574601968, 2.70961430055421932406734153548, 4.08047292942168149987346636493, 5.43604378190078653650128548413, 6.61069813333769198944009927879, 7.934971227553052378192820105367, 8.509081267817681463388950779282, 9.551201446942913015030291934024, 10.14242724505272250474643863845, 11.36557375328696613402873798062

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