Properties

Label 2-3330-1.1-c1-0-53
Degree $2$
Conductor $3330$
Sign $-1$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s − 3.37·7-s + 8-s − 10-s + 1.37·11-s + 1.37·13-s − 3.37·14-s + 16-s + 1.37·17-s − 1.37·19-s − 20-s + 1.37·22-s − 3.37·23-s + 25-s + 1.37·26-s − 3.37·28-s − 6·29-s + 2.74·31-s + 32-s + 1.37·34-s + 3.37·35-s − 37-s − 1.37·38-s − 40-s − 8.74·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.27·7-s + 0.353·8-s − 0.316·10-s + 0.413·11-s + 0.380·13-s − 0.901·14-s + 0.250·16-s + 0.332·17-s − 0.314·19-s − 0.223·20-s + 0.292·22-s − 0.703·23-s + 0.200·25-s + 0.269·26-s − 0.637·28-s − 1.11·29-s + 0.492·31-s + 0.176·32-s + 0.235·34-s + 0.570·35-s − 0.164·37-s − 0.222·38-s − 0.158·40-s − 1.36·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $-1$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
37 \( 1 + T \)
good7 \( 1 + 3.37T + 7T^{2} \)
11 \( 1 - 1.37T + 11T^{2} \)
13 \( 1 - 1.37T + 13T^{2} \)
17 \( 1 - 1.37T + 17T^{2} \)
19 \( 1 + 1.37T + 19T^{2} \)
23 \( 1 + 3.37T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 2.74T + 31T^{2} \)
41 \( 1 + 8.74T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 4.74T + 47T^{2} \)
53 \( 1 + 5.37T + 53T^{2} \)
59 \( 1 + 14.7T + 59T^{2} \)
61 \( 1 + 2.74T + 61T^{2} \)
67 \( 1 - 2.74T + 67T^{2} \)
71 \( 1 + 1.25T + 71T^{2} \)
73 \( 1 + 4.11T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 0.627T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−8.151112827566941020153314655912, −7.36716726103839439926041750527, −6.54859484058128516811337610090, −6.12314964877450324636714959788, −5.20782266792309859734974093443, −4.18185752622285749252271334572, −3.56666661532034164746079117739, −2.88675142955439155506145127817, −1.60753780957048935469755045442, 0, 1.60753780957048935469755045442, 2.88675142955439155506145127817, 3.56666661532034164746079117739, 4.18185752622285749252271334572, 5.20782266792309859734974093443, 6.12314964877450324636714959788, 6.54859484058128516811337610090, 7.36716726103839439926041750527, 8.151112827566941020153314655912

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