Properties

Label 2-3330-1.1-c1-0-27
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 + 3.68·7-s + 8-s − 10-s + 2.14·11-s − 1.48·13-s + 3.68·14-s + 16-s − 0.707·17-s − 1.48·19-s − 20-s + 2.14·22-s + 5.88·23-s + 25-s − 1.48·26-s + 3.68·28-s + 1.80·29-s + 10.0·31-s + 32-s − 0.707·34-s − 3.68·35-s − 37-s − 1.48·38-s − 40-s − 9.00·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.39·7-s + 0.353·8-s − 0.316·10-s + 0.647·11-s − 0.413·13-s + 0.985·14-s + 0.250·16-s − 0.171·17-s − 0.341·19-s − 0.223·20-s + 0.457·22-s + 1.22·23-s + 0.200·25-s − 0.292·26-s + 0.696·28-s + 0.334·29-s + 1.80·31-s + 0.176·32-s − 0.121·34-s − 0.623·35-s − 0.164·37-s − 0.241·38-s − 0.158·40-s − 1.40·41-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

\(L(1)\) \(\approx\) \(3.461849564\)
\(L(\frac12)\) \(\approx\) \(3.461849564\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.68T + 7T^{2} \)
11 \( 1 - 2.14T + 11T^{2} \)
13 \( 1 + 1.48T + 13T^{2} \)
17 \( 1 + 0.707T + 17T^{2} \)
19 \( 1 + 1.48T + 19T^{2} \)
23 \( 1 - 5.88T + 23T^{2} \)
29 \( 1 - 1.80T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
41 \( 1 + 9.00T + 41T^{2} \)
43 \( 1 - 2.36T + 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 + 5.51T + 53T^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 + 3.63T + 61T^{2} \)
67 \( 1 - 15.6T + 67T^{2} \)
71 \( 1 + 7.66T + 71T^{2} \)
73 \( 1 - 15.0T + 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 - 5.15T + 83T^{2} \)
89 \( 1 + 4.46T + 89T^{2} \)
97 \( 1 + 7.00T + 97T^{2} \)
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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

−8.229708880062826244512610517689, −8.107806407724230702169848724245, −6.89677858206849337570493242123, −6.56201587917051014614144666856, −5.23908378441880035264412197221, −4.83954014044841435874383885361, −4.11857067341460528949311028083, −3.14957058692443973291051858023, −2.11006660128168517913660186825, −1.07643003061251412209578338863, 1.07643003061251412209578338863, 2.11006660128168517913660186825, 3.14957058692443973291051858023, 4.11857067341460528949311028083, 4.83954014044841435874383885361, 5.23908378441880035264412197221, 6.56201587917051014614144666856, 6.89677858206849337570493242123, 8.107806407724230702169848724245, 8.229708880062826244512610517689

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