Properties

Label 2-3360-1.1-c1-0-31
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 + 4·11-s − 2·13-s + 15-s − 2·17-s − 4·19-s + 21-s + 25-s − 27-s + 10·29-s − 4·33-s + 35-s + 6·37-s + 2·39-s − 10·41-s − 8·43-s − 45-s − 4·47-s + 49-s + 2·51-s + 6·53-s − 4·55-s + 4·57-s + 10·61-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 − 0.917·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.696·33-s + 0.169·35-s + 0.986·37-s + 0.320·39-s − 1.56·41-s − 1.21·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s + 1.28·61-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: $\chi_{3360} (1, \cdot )$
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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−8.398488442216376060800585323034, −7.32412694070645715715841837301, −6.58231640570943695170243461055, −6.29210754378711639558402142799, −5.06313695660856529095288192450, −4.41505341237461983085321975339, −3.65314900404320595304089820888, −2.55858560243792014719014603355, −1.30012330121326572487096070830, 0, 1.30012330121326572487096070830, 2.55858560243792014719014603355, 3.65314900404320595304089820888, 4.41505341237461983085321975339, 5.06313695660856529095288192450, 6.29210754378711639558402142799, 6.58231640570943695170243461055, 7.32412694070645715715841837301, 8.398488442216376060800585323034

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