Properties

Label 2-3381-1.1-c1-0-102
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.36·2-s − 3-s − 0.135·4-s − 1.32·5-s − 1.36·6-s − 2.91·8-s + 9-s − 1.81·10-s − 1.42·11-s + 0.135·12-s + 3.16·13-s + 1.32·15-s − 3.71·16-s + 1.15·17-s + 1.36·18-s + 8.57·19-s + 0.179·20-s − 1.93·22-s − 23-s + 2.91·24-s − 3.24·25-s + 4.32·26-s − 27-s + 7.45·29-s + 1.81·30-s + 1.86·31-s + 0.764·32-s + ⋯
L(s)  = 1  + 0.965·2-s − 0.577·3-s − 0.0677·4-s − 0.593·5-s − 0.557·6-s − 1.03·8-s + 0.333·9-s − 0.572·10-s − 0.428·11-s + 0.0391·12-s + 0.878·13-s + 0.342·15-s − 0.927·16-s + 0.281·17-s + 0.321·18-s + 1.96·19-s + 0.0401·20-s − 0.413·22-s − 0.208·23-s + 0.595·24-s − 0.648·25-s + 0.847·26-s − 0.192·27-s + 1.38·29-s + 0.330·30-s + 0.335·31-s + 0.135·32-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: \(1\)
Selberg data: \((2,\ 3381,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
7 \( 1 \)
23 \( 1 + T \)
good2 \( 1 - 1.36T + 2T^{2} \)
5 \( 1 + 1.32T + 5T^{2} \)
11 \( 1 + 1.42T + 11T^{2} \)
13 \( 1 - 3.16T + 13T^{2} \)
17 \( 1 - 1.15T + 17T^{2} \)
19 \( 1 - 8.57T + 19T^{2} \)
29 \( 1 - 7.45T + 29T^{2} \)
31 \( 1 - 1.86T + 31T^{2} \)
37 \( 1 + 7.79T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 - 1.73T + 47T^{2} \)
53 \( 1 + 4.14T + 53T^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + 8.09T + 67T^{2} \)
71 \( 1 + 5.18T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 5.70T + 83T^{2} \)
89 \( 1 + 6.35T + 89T^{2} \)
97 \( 1 - 0.408T + 97T^{2} \)
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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

−8.241541897929178295297560463602, −7.35106705013664977040454550544, −6.55107093656804624199301626066, −5.74504967454925011672915123022, −5.14980874654942992831373337102, −4.49967637567644292453632006168, −3.50818803575482742990229973466, −3.08637088953727750778795066805, −1.37982863221226548384402762977, 0, 1.37982863221226548384402762977, 3.08637088953727750778795066805, 3.50818803575482742990229973466, 4.49967637567644292453632006168, 5.14980874654942992831373337102, 5.74504967454925011672915123022, 6.55107093656804624199301626066, 7.35106705013664977040454550544, 8.241541897929178295297560463602

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