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Properties

Label 2-2720-680.339-c0-0-4
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

\(L(\frac{1}{2})\) \(\approx\) \(1.511744652\)
\(L(\frac12)\) \(\approx\) \(1.511744652\)
\(L(1)\) 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

−8.744763443184652182534172704040, −8.356735175961630519588426590991, −7.63760242744065964660796513705, −7.03839187600605989078337306270, −5.90933423672003930063808663009, −5.07904411302221840784736581166, −3.86995060420898732126780616295, −3.46514720728198954626593092266, −2.61290429750629892385151107113, −1.16954522764747222840157357890, 1.16954522764747222840157357890, 2.61290429750629892385151107113, 3.46514720728198954626593092266, 3.86995060420898732126780616295, 5.07904411302221840784736581166, 5.90933423672003930063808663009, 7.03839187600605989078337306270, 7.63760242744065964660796513705, 8.356735175961630519588426590991, 8.744763443184652182534172704040

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