Properties

Label 2-480-1.1-c1-0-1
Degree $2$
Conductor $480$
Sign $1$
Analytic cond. $3.83281$
Root an. cond. $1.95775$
Motivic weight $1$
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 + 9-s + 2·13-s − 15-s + 6·17-s − 4·19-s + 8·23-s + 25-s − 27-s − 2·29-s + 4·31-s + 10·37-s − 2·39-s + 2·41-s − 4·43-s + 45-s + 8·47-s − 7·49-s − 6·51-s − 2·53-s + 4·57-s + 8·59-s − 2·61-s + 2·65-s − 12·67-s − 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 1.64·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s − 49-s − 0.840·51-s − 0.274·53-s + 0.529·57-s + 1.04·59-s − 0.256·61-s + 0.248·65-s − 1.46·67-s − 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(3.83281\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{480} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.341664838\)
\(L(\frac12)\) \(\approx\) \(1.341664838\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−10.97163175619009206137205774068, −10.19227653538760520507936434046, −9.328467546447739299100058924123, −8.304308599781223866728910598449, −7.22612323207528452617340244852, −6.22207440469416781281899599638, −5.45564722344211377728395826291, −4.33436868576345517333214359577, −2.93279961256840981884560530804, −1.21912253617593317626144578411, 1.21912253617593317626144578411, 2.93279961256840981884560530804, 4.33436868576345517333214359577, 5.45564722344211377728395826291, 6.22207440469416781281899599638, 7.22612323207528452617340244852, 8.304308599781223866728910598449, 9.328467546447739299100058924123, 10.19227653538760520507936434046, 10.97163175619009206137205774068

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