Properties

Label 2-3330-1.1-c1-0-19
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 − 0.0586·7-s + 8-s − 10-s − 1.24·11-s + 3.77·13-s − 0.0586·14-s + 16-s − 7.49·17-s + 3.77·19-s − 20-s − 1.24·22-s + 3.66·23-s + 25-s + 3.77·26-s − 0.0586·28-s + 0.280·29-s + 4.41·31-s + 32-s − 7.49·34-s + 0.0586·35-s − 37-s + 3.77·38-s − 40-s + 7.14·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.0221·7-s + 0.353·8-s − 0.316·10-s − 0.376·11-s + 1.04·13-s − 0.0156·14-s + 0.250·16-s − 1.81·17-s + 0.866·19-s − 0.223·20-s − 0.266·22-s + 0.763·23-s + 0.200·25-s + 0.741·26-s − 0.0110·28-s + 0.0520·29-s + 0.792·31-s + 0.176·32-s − 1.28·34-s + 0.00991·35-s − 0.164·37-s + 0.612·38-s − 0.158·40-s + 1.11·41-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: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.764130346\)
\(L(\frac12)\) \(\approx\) \(2.764130346\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.0586T + 7T^{2} \)
11 \( 1 + 1.24T + 11T^{2} \)
13 \( 1 - 3.77T + 13T^{2} \)
17 \( 1 + 7.49T + 17T^{2} \)
19 \( 1 - 3.77T + 19T^{2} \)
23 \( 1 - 3.66T + 23T^{2} \)
29 \( 1 - 0.280T + 29T^{2} \)
31 \( 1 - 4.41T + 31T^{2} \)
41 \( 1 - 7.14T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 - 6.05T + 47T^{2} \)
53 \( 1 - 5.36T + 53T^{2} \)
59 \( 1 + 0.615T + 59T^{2} \)
61 \( 1 - 5.02T + 61T^{2} \)
67 \( 1 - 1.38T + 67T^{2} \)
71 \( 1 - 6.61T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 - 5.50T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 - 9.14T + 97T^{2} \)
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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

−8.650699870898007363139774530559, −7.74583059327674524177431563366, −7.06107639942601007484643409050, −6.32147799687527701090554089413, −5.58741157089753680495152057474, −4.64121054259790498806912908127, −4.06978361203335588335476122786, −3.12087774864937357790907931873, −2.28690693960173524393266366471, −0.907152712227820404555887963628, 0.907152712227820404555887963628, 2.28690693960173524393266366471, 3.12087774864937357790907931873, 4.06978361203335588335476122786, 4.64121054259790498806912908127, 5.58741157089753680495152057474, 6.32147799687527701090554089413, 7.06107639942601007484643409050, 7.74583059327674524177431563366, 8.650699870898007363139774530559

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