Properties

Label 2-3360-1.1-c1-0-10
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 7-s + 9-s + 4·11-s − 2·13-s − 15-s − 2·17-s + 8·19-s + 21-s + 25-s − 27-s − 2·29-s − 4·33-s − 35-s − 6·37-s + 2·39-s + 2·41-s + 4·43-s + 45-s + 8·47-s + 49-s + 2·51-s − 6·53-s + 4·55-s − 8·57-s − 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s + 1.83·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s − 0.169·35-s − 0.986·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.539·55-s − 1.05·57-s − 0.256·61-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: \(0\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.682598714\)
\(L(\frac12)\) \(\approx\) \(1.682598714\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−8.890905071405940263807660308258, −7.64478753337256159133360922574, −7.06086036239019688063680452315, −6.35316640417836655320198380518, −5.63632782549750996248123136110, −4.91380017540757573217630253175, −3.97110146144033917413647839403, −3.09040679008008900975048055760, −1.89425390748404433456997372556, −0.818727734574118390759677993169, 0.818727734574118390759677993169, 1.89425390748404433456997372556, 3.09040679008008900975048055760, 3.97110146144033917413647839403, 4.91380017540757573217630253175, 5.63632782549750996248123136110, 6.35316640417836655320198380518, 7.06086036239019688063680452315, 7.64478753337256159133360922574, 8.890905071405940263807660308258

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