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Properties

Label 2-2720-680.339-c0-0-1
Degree $2$
Conductor $2720$
Sign $1$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 13-s + 15-s − 17-s + 19-s + 25-s + 27-s + 29-s − 31-s + 39-s + 47-s + 49-s + 51-s − 53-s − 57-s + 59-s + 61-s + 65-s − 71-s + 73-s − 75-s + 2·79-s − 81-s + 85-s − 87-s − 89-s + ⋯
L(s)  = 1  − 3-s − 5-s − 13-s + 15-s − 17-s + 19-s + 25-s + 27-s + 29-s − 31-s + 39-s + 47-s + 49-s + 51-s − 53-s − 57-s + 59-s + 61-s + 65-s − 71-s + 73-s − 75-s + 2·79-s − 81-s + 85-s − 87-s − 89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $1$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2720} (1359, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2720,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5398218118\)
\(L(\frac12)\) \(\approx\) \(0.5398218118\)
\(L(1)\) 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 \( 1 + T \)
17 \( 1 + T \)
good3 \( 1 + T + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 - T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 - T + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( ( 1 - T )^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T + T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.953721984378013146451965959053, −8.226838368981026667541897349792, −7.29616420060564575802186873079, −6.87284828746794899678553210822, −5.87501318340168522650533382157, −5.06114665322043199205552353887, −4.49279357154404013214470673763, −3.44892238198779535897256672260, −2.40794142918273952362710800015, −0.68834452274414551487474705001, 0.68834452274414551487474705001, 2.40794142918273952362710800015, 3.44892238198779535897256672260, 4.49279357154404013214470673763, 5.06114665322043199205552353887, 5.87501318340168522650533382157, 6.87284828746794899678553210822, 7.29616420060564575802186873079, 8.226838368981026667541897349792, 8.953721984378013146451965959053

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