Properties

Label 2-3330-1.1-c1-0-36
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 4.56·7-s + 8-s + 10-s − 3.41·11-s + 0.971·13-s + 4.56·14-s + 16-s − 2.56·17-s + 4.86·19-s + 20-s − 3.41·22-s + 1.02·23-s + 25-s + 0.971·26-s + 4.56·28-s − 3.59·29-s + 2.55·31-s + 32-s − 2.56·34-s + 4.56·35-s + 37-s + 4.86·38-s + 40-s + 7.74·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.72·7-s + 0.353·8-s + 0.316·10-s − 1.03·11-s + 0.269·13-s + 1.22·14-s + 0.250·16-s − 0.622·17-s + 1.11·19-s + 0.223·20-s − 0.728·22-s + 0.214·23-s + 0.200·25-s + 0.190·26-s + 0.863·28-s − 0.667·29-s + 0.458·31-s + 0.176·32-s − 0.440·34-s + 0.772·35-s + 0.164·37-s + 0.789·38-s + 0.158·40-s + 1.20·41-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: $\chi_{3330} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.995415292\)
\(L(\frac12)\) \(\approx\) \(3.995415292\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.56T + 7T^{2} \)
11 \( 1 + 3.41T + 11T^{2} \)
13 \( 1 - 0.971T + 13T^{2} \)
17 \( 1 + 2.56T + 17T^{2} \)
19 \( 1 - 4.86T + 19T^{2} \)
23 \( 1 - 1.02T + 23T^{2} \)
29 \( 1 + 3.59T + 29T^{2} \)
31 \( 1 - 2.55T + 31T^{2} \)
41 \( 1 - 7.74T + 41T^{2} \)
43 \( 1 - 9.74T + 43T^{2} \)
47 \( 1 - 4.99T + 47T^{2} \)
53 \( 1 + 6.71T + 53T^{2} \)
59 \( 1 - 6.83T + 59T^{2} \)
61 \( 1 + 6.99T + 61T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 + 8.29T + 71T^{2} \)
73 \( 1 - 1.56T + 73T^{2} \)
79 \( 1 + 6.29T + 79T^{2} \)
83 \( 1 - 7.86T + 83T^{2} \)
89 \( 1 + 17.3T + 89T^{2} \)
97 \( 1 + 0.747T + 97T^{2} \)
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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

−8.544025536703326258670891720799, −7.57513071058190581004678604830, −7.43470489998096180473278523431, −6.05511715322896558440639280924, −5.52294127835059554379639219806, −4.80061832045932372725559662541, −4.22515266862683460625669271459, −2.93353842991766621689934137868, −2.15604879615561611124697767487, −1.17974416425608377846294685597, 1.17974416425608377846294685597, 2.15604879615561611124697767487, 2.93353842991766621689934137868, 4.22515266862683460625669271459, 4.80061832045932372725559662541, 5.52294127835059554379639219806, 6.05511715322896558440639280924, 7.43470489998096180473278523431, 7.57513071058190581004678604830, 8.544025536703326258670891720799

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