Properties

Label 2-3381-1.1-c1-0-149
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  + 2.63·2-s − 3-s + 4.95·4-s − 0.742·5-s − 2.63·6-s + 7.80·8-s + 9-s − 1.95·10-s − 5.37·11-s − 4.95·12-s − 2.10·13-s + 0.742·15-s + 10.6·16-s − 7.85·17-s + 2.63·18-s − 7.61·19-s − 3.68·20-s − 14.1·22-s − 23-s − 7.80·24-s − 4.44·25-s − 5.54·26-s − 27-s − 2.52·29-s + 1.95·30-s + 7.75·31-s + 12.5·32-s + ⋯
L(s)  = 1  + 1.86·2-s − 0.577·3-s + 2.47·4-s − 0.332·5-s − 1.07·6-s + 2.75·8-s + 0.333·9-s − 0.619·10-s − 1.62·11-s − 1.43·12-s − 0.582·13-s + 0.191·15-s + 2.66·16-s − 1.90·17-s + 0.621·18-s − 1.74·19-s − 0.822·20-s − 3.02·22-s − 0.208·23-s − 1.59·24-s − 0.889·25-s − 1.08·26-s − 0.192·27-s − 0.468·29-s + 0.357·30-s + 1.39·31-s + 2.21·32-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: Trivial
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 \( 1 - 2.63T + 2T^{2} \)
5 \( 1 + 0.742T + 5T^{2} \)
11 \( 1 + 5.37T + 11T^{2} \)
13 \( 1 + 2.10T + 13T^{2} \)
17 \( 1 + 7.85T + 17T^{2} \)
19 \( 1 + 7.61T + 19T^{2} \)
29 \( 1 + 2.52T + 29T^{2} \)
31 \( 1 - 7.75T + 31T^{2} \)
37 \( 1 + 7.43T + 37T^{2} \)
41 \( 1 - 7.59T + 41T^{2} \)
43 \( 1 - 7.04T + 43T^{2} \)
47 \( 1 - 5.73T + 47T^{2} \)
53 \( 1 - 6.84T + 53T^{2} \)
59 \( 1 + 1.30T + 59T^{2} \)
61 \( 1 + 3.10T + 61T^{2} \)
67 \( 1 + 9.87T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 1.23T + 73T^{2} \)
79 \( 1 - 0.596T + 79T^{2} \)
83 \( 1 - 0.194T + 83T^{2} \)
89 \( 1 + 1.61T + 89T^{2} \)
97 \( 1 - 9.67T + 97T^{2} \)
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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

−7.84030198423162685615288946026, −7.21903107912073190010168765132, −6.41123778161367642889769784664, −5.87995023718196255350016686623, −5.00809209191582257930113896059, −4.46655703407219142971292990234, −3.88802094877354311535115451669, −2.48740884125946952075983389617, −2.23791278365569032886392594517, 0, 2.23791278365569032886392594517, 2.48740884125946952075983389617, 3.88802094877354311535115451669, 4.46655703407219142971292990234, 5.00809209191582257930113896059, 5.87995023718196255350016686623, 6.41123778161367642889769784664, 7.21903107912073190010168765132, 7.84030198423162685615288946026

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