Properties

Label 2-3360-1.1-c1-0-17
Degree $2$
Conductor $3360$
Sign $1$
Analytic cond. $26.8297$
Root an. cond. $5.17974$
Motivic weight $1$
Arithmetic yes
Rational no
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 − 4.47·13-s + 15-s − 4.47·17-s + 21-s + 6.47·23-s + 25-s + 27-s + 8.47·29-s − 6.47·31-s + 35-s + 8.47·37-s − 4.47·39-s + 10.9·41-s + 10.4·43-s + 45-s − 2.47·47-s + 49-s − 4.47·51-s + 2·53-s − 4·59-s + 12.4·61-s + 63-s − 4.47·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s − 1.24·13-s + 0.258·15-s − 1.08·17-s + 0.218·21-s + 1.34·23-s + 0.200·25-s + 0.192·27-s + 1.57·29-s − 1.16·31-s + 0.169·35-s + 1.39·37-s − 0.716·39-s + 1.70·41-s + 1.59·43-s + 0.149·45-s − 0.360·47-s + 0.142·49-s − 0.626·51-s + 0.274·53-s − 0.520·59-s + 1.59·61-s + 0.125·63-s − 0.554·65-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: no
Arithmetic: yes
Character: $\chi_{3360} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.594638158\)
\(L(\frac12)\) \(\approx\) \(2.594638158\)
\(L(\frac{3}{2})\) 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 \( 1 + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 6.47T + 23T^{2} \)
29 \( 1 - 8.47T + 29T^{2} \)
31 \( 1 + 6.47T + 31T^{2} \)
37 \( 1 - 8.47T + 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + 2.47T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 - 2.47T + 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 - 8.47T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + 6.94T + 89T^{2} \)
97 \( 1 + 4.47T + 97T^{2} \)
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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

−8.667615124910568525897941137695, −7.894279060320489019889263271089, −7.15824262205076958805389537432, −6.54695511885551053288807143244, −5.47678973147421015334903150988, −4.74160312367346261047832282929, −4.04236370906893394626285276808, −2.68579138043797739454868308621, −2.33493158427018902618279765864, −0.952004215443621182430233406141, 0.952004215443621182430233406141, 2.33493158427018902618279765864, 2.68579138043797739454868308621, 4.04236370906893394626285276808, 4.74160312367346261047832282929, 5.47678973147421015334903150988, 6.54695511885551053288807143244, 7.15824262205076958805389537432, 7.894279060320489019889263271089, 8.667615124910568525897941137695

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