Properties

Label 2-3381-1.1-c1-0-26
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  + 0.533·2-s + 3-s − 1.71·4-s − 1.09·5-s + 0.533·6-s − 1.98·8-s + 9-s − 0.583·10-s − 3.05·11-s − 1.71·12-s − 4.92·13-s − 1.09·15-s + 2.37·16-s + 0.748·17-s + 0.533·18-s + 1.24·19-s + 1.87·20-s − 1.63·22-s − 23-s − 1.98·24-s − 3.80·25-s − 2.62·26-s + 27-s + 4.90·29-s − 0.583·30-s + 8.55·31-s + 5.23·32-s + ⋯
L(s)  = 1  + 0.377·2-s + 0.577·3-s − 0.857·4-s − 0.489·5-s + 0.217·6-s − 0.701·8-s + 0.333·9-s − 0.184·10-s − 0.921·11-s − 0.495·12-s − 1.36·13-s − 0.282·15-s + 0.592·16-s + 0.181·17-s + 0.125·18-s + 0.284·19-s + 0.419·20-s − 0.347·22-s − 0.208·23-s − 0.404·24-s − 0.760·25-s − 0.515·26-s + 0.192·27-s + 0.910·29-s − 0.106·30-s + 1.53·31-s + 0.924·32-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: \(0\)
Selberg data: \((2,\ 3381,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.457431368\)
\(L(\frac12)\) \(\approx\) \(1.457431368\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.533T + 2T^{2} \)
5 \( 1 + 1.09T + 5T^{2} \)
11 \( 1 + 3.05T + 11T^{2} \)
13 \( 1 + 4.92T + 13T^{2} \)
17 \( 1 - 0.748T + 17T^{2} \)
19 \( 1 - 1.24T + 19T^{2} \)
29 \( 1 - 4.90T + 29T^{2} \)
31 \( 1 - 8.55T + 31T^{2} \)
37 \( 1 - 5.89T + 37T^{2} \)
41 \( 1 + 3.90T + 41T^{2} \)
43 \( 1 + 2.45T + 43T^{2} \)
47 \( 1 - 5.39T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 - 4.12T + 59T^{2} \)
61 \( 1 + 8.22T + 61T^{2} \)
67 \( 1 - 3.21T + 67T^{2} \)
71 \( 1 + 5.13T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 - 2.24T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 - 6.67T + 97T^{2} \)
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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

−8.476906499184357520425632858720, −7.929882340939475097301420513487, −7.38581638829660892190599362673, −6.30547278499939884074486067466, −5.31281206821198389478453191899, −4.71659651130715055173905489280, −4.02707109178584770451008582728, −3.06399701808452138095063257058, −2.37394240642366791808495220936, −0.64154992193545169012728136860, 0.64154992193545169012728136860, 2.37394240642366791808495220936, 3.06399701808452138095063257058, 4.02707109178584770451008582728, 4.71659651130715055173905489280, 5.31281206821198389478453191899, 6.30547278499939884074486067466, 7.38581638829660892190599362673, 7.929882340939475097301420513487, 8.476906499184357520425632858720

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