Properties

Label 2-3330-1.1-c1-0-11
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 − 1.19·7-s − 8-s + 10-s + 3.64·11-s + 4.86·13-s + 1.19·14-s + 16-s − 3.19·17-s + 2.27·19-s − 20-s − 3.64·22-s + 2.86·23-s + 25-s − 4.86·26-s − 1.19·28-s − 6.06·29-s + 5.36·31-s − 32-s + 3.19·34-s + 1.19·35-s + 37-s − 2.27·38-s + 40-s + 8.75·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.450·7-s − 0.353·8-s + 0.316·10-s + 1.09·11-s + 1.35·13-s + 0.318·14-s + 0.250·16-s − 0.774·17-s + 0.521·19-s − 0.223·20-s − 0.776·22-s + 0.598·23-s + 0.200·25-s − 0.954·26-s − 0.225·28-s − 1.12·29-s + 0.963·31-s − 0.176·32-s + 0.547·34-s + 0.201·35-s + 0.164·37-s − 0.369·38-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: $\chi_{3330} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.252678769\)
\(L(\frac12)\) \(\approx\) \(1.252678769\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.19T + 7T^{2} \)
11 \( 1 - 3.64T + 11T^{2} \)
13 \( 1 - 4.86T + 13T^{2} \)
17 \( 1 + 3.19T + 17T^{2} \)
19 \( 1 - 2.27T + 19T^{2} \)
23 \( 1 - 2.86T + 23T^{2} \)
29 \( 1 + 6.06T + 29T^{2} \)
31 \( 1 - 5.36T + 31T^{2} \)
41 \( 1 - 8.75T + 41T^{2} \)
43 \( 1 + 6.75T + 43T^{2} \)
47 \( 1 + 4.13T + 47T^{2} \)
53 \( 1 + 5.88T + 53T^{2} \)
59 \( 1 + 7.28T + 59T^{2} \)
61 \( 1 - 14.5T + 61T^{2} \)
67 \( 1 + 4.90T + 67T^{2} \)
71 \( 1 + 5.38T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 - 7.38T + 79T^{2} \)
83 \( 1 + 4.41T + 83T^{2} \)
89 \( 1 + 9.63T + 89T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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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

−8.678477192735905801799787117619, −8.032654015623805802776434422897, −7.13114392831936220113820293080, −6.49015156191513113367381977847, −5.93673125883930980215055182745, −4.69358193313141894089764923850, −3.75697525328792306958560876014, −3.11970214206761240646152488649, −1.77668608694996462372195969161, −0.76905522154442237512605290149, 0.76905522154442237512605290149, 1.77668608694996462372195969161, 3.11970214206761240646152488649, 3.75697525328792306958560876014, 4.69358193313141894089764923850, 5.93673125883930980215055182745, 6.49015156191513113367381977847, 7.13114392831936220113820293080, 8.032654015623805802776434422897, 8.678477192735905801799787117619

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