Properties

Label 2-3360-1.1-c1-0-28
Degree $2$
Conductor $3360$
Sign $-1$
Analytic cond. $26.8297$
Root an. cond. $5.17974$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s + 9-s − 2·11-s + 15-s − 6·17-s + 6·19-s + 21-s + 8·23-s + 25-s − 27-s + 6·29-s + 6·31-s + 2·33-s + 35-s − 10·37-s + 2·41-s + 4·43-s − 45-s − 8·47-s + 49-s + 6·51-s − 12·53-s + 2·55-s − 6·57-s − 12·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.258·15-s − 1.45·17-s + 1.37·19-s + 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.348·33-s + 0.169·35-s − 1.64·37-s + 0.312·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.840·51-s − 1.64·53-s + 0.269·55-s − 0.794·57-s − 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(26.8297\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−8.220445638160044929496837312007, −7.37497321130478871197915841208, −6.76360727272757381454020390912, −6.09270049536733414349274513765, −4.92025339438886395369489047888, −4.72916901985484185209248139884, −3.39896233277134585610990366575, −2.71557781749145545205368216765, −1.24960858885676340498672394499, 0, 1.24960858885676340498672394499, 2.71557781749145545205368216765, 3.39896233277134585610990366575, 4.72916901985484185209248139884, 4.92025339438886395369489047888, 6.09270049536733414349274513765, 6.76360727272757381454020390912, 7.37497321130478871197915841208, 8.220445638160044929496837312007

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