Properties

Label 2-3360-1.1-c1-0-9
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 + 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: \(0\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.001049487\)
\(L(\frac12)\) \(\approx\) \(2.001049487\)
\(L(\frac{3}{2})\) 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

−8.437871548224310202742352185353, −7.949152959733991878916874270549, −7.31979595951381878058034312516, −6.55341822008387752471814319456, −5.36060461079758519453871924476, −4.81491928124302049005015756014, −3.92742709962574332759519972128, −2.88633232471614386188270893470, −2.28839099576648481139527420277, −0.813170101248483559516930675379, 0.813170101248483559516930675379, 2.28839099576648481139527420277, 2.88633232471614386188270893470, 3.92742709962574332759519972128, 4.81491928124302049005015756014, 5.36060461079758519453871924476, 6.55341822008387752471814319456, 7.31979595951381878058034312516, 7.949152959733991878916874270549, 8.437871548224310202742352185353

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