Properties

Label 2-3381-1.1-c1-0-4
Degree $2$
Conductor $3381$
Sign $1$
Analytic cond. $26.9974$
Root an. cond. $5.19590$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.73·2-s − 3-s + 5.50·4-s − 3.38·5-s + 2.73·6-s − 9.61·8-s + 9-s + 9.26·10-s + 1.88·11-s − 5.50·12-s + 1.76·13-s + 3.38·15-s + 15.3·16-s − 5.89·17-s − 2.73·18-s − 6.62·19-s − 18.6·20-s − 5.16·22-s + 23-s + 9.61·24-s + 6.43·25-s − 4.84·26-s − 27-s + 8.24·29-s − 9.26·30-s + 2.94·31-s − 22.7·32-s + ⋯
L(s)  = 1  − 1.93·2-s − 0.577·3-s + 2.75·4-s − 1.51·5-s + 1.11·6-s − 3.39·8-s + 0.333·9-s + 2.93·10-s + 0.567·11-s − 1.58·12-s + 0.490·13-s + 0.873·15-s + 3.82·16-s − 1.42·17-s − 0.645·18-s − 1.52·19-s − 4.16·20-s − 1.10·22-s + 0.208·23-s + 1.96·24-s + 1.28·25-s − 0.950·26-s − 0.192·27-s + 1.53·29-s − 1.69·30-s + 0.528·31-s − 4.02·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(26.9974\)
Root analytic conductor: \(5.19590\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2204512213\)
\(L(\frac12)\) \(\approx\) \(0.2204512213\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 \)
23 \( 1 - T \)
good2 \( 1 + 2.73T + 2T^{2} \)
5 \( 1 + 3.38T + 5T^{2} \)
11 \( 1 - 1.88T + 11T^{2} \)
13 \( 1 - 1.76T + 13T^{2} \)
17 \( 1 + 5.89T + 17T^{2} \)
19 \( 1 + 6.62T + 19T^{2} \)
29 \( 1 - 8.24T + 29T^{2} \)
31 \( 1 - 2.94T + 31T^{2} \)
37 \( 1 + 4.43T + 37T^{2} \)
41 \( 1 + 5.90T + 41T^{2} \)
43 \( 1 - 0.669T + 43T^{2} \)
47 \( 1 - 1.79T + 47T^{2} \)
53 \( 1 + 9.68T + 53T^{2} \)
59 \( 1 - 1.36T + 59T^{2} \)
61 \( 1 - 9.36T + 61T^{2} \)
67 \( 1 + 8.17T + 67T^{2} \)
71 \( 1 + 0.295T + 71T^{2} \)
73 \( 1 + 4.45T + 73T^{2} \)
79 \( 1 + 3.02T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.585963158331735619003293783371, −8.148798702047248372750663148425, −7.21834657040902517983387883389, −6.64932810755966003178867907357, −6.23100932261127160950241437553, −4.68436725343949250436231816429, −3.83291593272185651269079036050, −2.70470802211457488615488536910, −1.52915434460346669921192168325, −0.39250097494924073764365974975, 0.39250097494924073764365974975, 1.52915434460346669921192168325, 2.70470802211457488615488536910, 3.83291593272185651269079036050, 4.68436725343949250436231816429, 6.23100932261127160950241437553, 6.64932810755966003178867907357, 7.21834657040902517983387883389, 8.148798702047248372750663148425, 8.585963158331735619003293783371

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