Properties

Label 2-3381-1.1-c1-0-108
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.51·2-s − 3-s + 4.32·4-s + 4.31·5-s − 2.51·6-s + 5.85·8-s + 9-s + 10.8·10-s + 1.72·11-s − 4.32·12-s − 2.88·13-s − 4.31·15-s + 6.06·16-s − 4.93·17-s + 2.51·18-s + 2.44·19-s + 18.6·20-s + 4.34·22-s + 23-s − 5.85·24-s + 13.6·25-s − 7.26·26-s − 27-s + 8.05·29-s − 10.8·30-s + 0.915·31-s + 3.55·32-s + ⋯
L(s)  = 1  + 1.77·2-s − 0.577·3-s + 2.16·4-s + 1.93·5-s − 1.02·6-s + 2.06·8-s + 0.333·9-s + 3.43·10-s + 0.521·11-s − 1.24·12-s − 0.801·13-s − 1.11·15-s + 1.51·16-s − 1.19·17-s + 0.592·18-s + 0.560·19-s + 4.17·20-s + 0.927·22-s + 0.208·23-s − 1.19·24-s + 2.72·25-s − 1.42·26-s − 0.192·27-s + 1.49·29-s − 1.98·30-s + 0.164·31-s + 0.627·32-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

\(L(1)\) \(\approx\) \(6.810854019\)
\(L(\frac12)\) \(\approx\) \(6.810854019\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.51T + 2T^{2} \)
5 \( 1 - 4.31T + 5T^{2} \)
11 \( 1 - 1.72T + 11T^{2} \)
13 \( 1 + 2.88T + 13T^{2} \)
17 \( 1 + 4.93T + 17T^{2} \)
19 \( 1 - 2.44T + 19T^{2} \)
29 \( 1 - 8.05T + 29T^{2} \)
31 \( 1 - 0.915T + 31T^{2} \)
37 \( 1 + 9.15T + 37T^{2} \)
41 \( 1 + 6.60T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + 0.399T + 47T^{2} \)
53 \( 1 - 6.54T + 53T^{2} \)
59 \( 1 + 1.83T + 59T^{2} \)
61 \( 1 + 1.30T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 9.00T + 71T^{2} \)
73 \( 1 + 2.61T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 3.07T + 97T^{2} \)
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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

−8.792868652059625616700977695036, −7.23283702121188838849990419651, −6.65977530580639820002307665617, −6.21252707657880461025495909652, −5.43770093704832849002524306103, −4.96891250707775053350463209396, −4.27051434405758784388727434432, −2.98532775087497435752267071806, −2.30871749165608807333858752406, −1.43188419819844126745875869213, 1.43188419819844126745875869213, 2.30871749165608807333858752406, 2.98532775087497435752267071806, 4.27051434405758784388727434432, 4.96891250707775053350463209396, 5.43770093704832849002524306103, 6.21252707657880461025495909652, 6.65977530580639820002307665617, 7.23283702121188838849990419651, 8.792868652059625616700977695036

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