Properties

Label 2-3381-1.1-c1-0-62
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  + 2.27·2-s − 3-s + 3.17·4-s − 1.23·5-s − 2.27·6-s + 2.67·8-s + 9-s − 2.81·10-s + 6.12·11-s − 3.17·12-s + 0.910·13-s + 1.23·15-s − 0.266·16-s + 4.90·17-s + 2.27·18-s − 6.90·19-s − 3.93·20-s + 13.9·22-s + 23-s − 2.67·24-s − 3.46·25-s + 2.07·26-s − 27-s + 3.36·29-s + 2.81·30-s + 5.43·31-s − 5.95·32-s + ⋯
L(s)  = 1  + 1.60·2-s − 0.577·3-s + 1.58·4-s − 0.553·5-s − 0.928·6-s + 0.945·8-s + 0.333·9-s − 0.891·10-s + 1.84·11-s − 0.916·12-s + 0.252·13-s + 0.319·15-s − 0.0665·16-s + 1.19·17-s + 0.536·18-s − 1.58·19-s − 0.879·20-s + 2.97·22-s + 0.208·23-s − 0.545·24-s − 0.693·25-s + 0.406·26-s − 0.192·27-s + 0.623·29-s + 0.514·30-s + 0.976·31-s − 1.05·32-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

\(L(1)\) \(\approx\) \(4.033202544\)
\(L(\frac12)\) \(\approx\) \(4.033202544\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.27T + 2T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
11 \( 1 - 6.12T + 11T^{2} \)
13 \( 1 - 0.910T + 13T^{2} \)
17 \( 1 - 4.90T + 17T^{2} \)
19 \( 1 + 6.90T + 19T^{2} \)
29 \( 1 - 3.36T + 29T^{2} \)
31 \( 1 - 5.43T + 31T^{2} \)
37 \( 1 - 6.54T + 37T^{2} \)
41 \( 1 + 2.52T + 41T^{2} \)
43 \( 1 + 0.231T + 43T^{2} \)
47 \( 1 - 13.3T + 47T^{2} \)
53 \( 1 - 4.22T + 53T^{2} \)
59 \( 1 + 0.754T + 59T^{2} \)
61 \( 1 - 3.57T + 61T^{2} \)
67 \( 1 + 4.49T + 67T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 + 4.27T + 73T^{2} \)
79 \( 1 + 0.222T + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 - 0.559T + 89T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−8.520503291085039161279938724584, −7.57380301983181149331789478660, −6.63388757470203762110831101716, −6.30003123675876328905280795247, −5.58382381124010398218864961123, −4.56927291537814256740349668487, −4.05867177391570342324124699627, −3.52228352529294159507565825795, −2.30523363321129628726147907711, −1.01671477517800375864409363671, 1.01671477517800375864409363671, 2.30523363321129628726147907711, 3.52228352529294159507565825795, 4.05867177391570342324124699627, 4.56927291537814256740349668487, 5.58382381124010398218864961123, 6.30003123675876328905280795247, 6.63388757470203762110831101716, 7.57380301983181149331789478660, 8.520503291085039161279938724584

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