Properties

Label 2-3330-1.1-c1-0-41
Degree $2$
Conductor $3330$
Sign $-1$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 5·7-s − 8-s − 10-s + 5·11-s − 13-s + 5·14-s + 16-s + 5·17-s − 3·19-s + 20-s − 5·22-s − 3·23-s + 25-s + 26-s − 5·28-s − 6·29-s − 6·31-s − 32-s − 5·34-s − 5·35-s − 37-s + 3·38-s − 40-s + 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.88·7-s − 0.353·8-s − 0.316·10-s + 1.50·11-s − 0.277·13-s + 1.33·14-s + 1/4·16-s + 1.21·17-s − 0.688·19-s + 0.223·20-s − 1.06·22-s − 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.944·28-s − 1.11·29-s − 1.07·31-s − 0.176·32-s − 0.857·34-s − 0.845·35-s − 0.164·37-s + 0.486·38-s − 0.158·40-s + 0.609·43-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: yes
Arithmetic: yes
Character: $\chi_{3330} (1, \cdot )$
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 \( 1 + 5 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−8.506167921955232129928080315022, −7.26238630113822102791797748853, −6.93245784370521961612621544079, −6.00025851812158853517523212456, −5.70716897529375252249064677689, −4.02906718830663741743009133501, −3.47738567939455373126598827256, −2.47043245971722245325321389426, −1.33411491837942943308142579279, 0, 1.33411491837942943308142579279, 2.47043245971722245325321389426, 3.47738567939455373126598827256, 4.02906718830663741743009133501, 5.70716897529375252249064677689, 6.00025851812158853517523212456, 6.93245784370521961612621544079, 7.26238630113822102791797748853, 8.506167921955232129928080315022

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