Properties

Label 2-3360-1.1-c1-0-37
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 − 4·13-s − 15-s − 2·17-s + 6·19-s − 21-s + 4·23-s + 25-s − 27-s − 10·29-s − 2·31-s + 2·33-s + 35-s − 2·37-s + 4·39-s − 10·41-s − 4·43-s + 45-s + 8·47-s + 49-s + 2·51-s + 4·53-s − 2·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.258·15-s − 0.485·17-s + 1.37·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.359·31-s + 0.348·33-s + 0.169·35-s − 0.328·37-s + 0.640·39-s − 1.56·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.549·53-s − 0.269·55-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−8.156950132487524347768697988630, −7.28275321982778119523411428494, −6.97439595419553187043159101361, −5.71407639949149005795291457738, −5.29253341744028685769995216301, −4.65017319605294256719503522655, −3.47052355220114510674535662547, −2.44806991329261649067716795810, −1.46527480814571706056391586293, 0, 1.46527480814571706056391586293, 2.44806991329261649067716795810, 3.47052355220114510674535662547, 4.65017319605294256719503522655, 5.29253341744028685769995216301, 5.71407639949149005795291457738, 6.97439595419553187043159101361, 7.28275321982778119523411428494, 8.156950132487524347768697988630

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