Properties

Label 2-3381-1.1-c1-0-29
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  − 2.06·2-s − 3-s + 2.26·4-s − 0.608·5-s + 2.06·6-s − 0.546·8-s + 9-s + 1.25·10-s − 1.10·11-s − 2.26·12-s + 1.84·13-s + 0.608·15-s − 3.40·16-s + 5.88·17-s − 2.06·18-s + 1.08·19-s − 1.37·20-s + 2.27·22-s + 23-s + 0.546·24-s − 4.62·25-s − 3.81·26-s − 27-s − 0.804·29-s − 1.25·30-s + 8.77·31-s + 8.11·32-s + ⋯
L(s)  = 1  − 1.46·2-s − 0.577·3-s + 1.13·4-s − 0.272·5-s + 0.843·6-s − 0.193·8-s + 0.333·9-s + 0.397·10-s − 0.332·11-s − 0.653·12-s + 0.512·13-s + 0.157·15-s − 0.850·16-s + 1.42·17-s − 0.486·18-s + 0.249·19-s − 0.308·20-s + 0.485·22-s + 0.208·23-s + 0.111·24-s − 0.925·25-s − 0.748·26-s − 0.192·27-s − 0.149·29-s − 0.229·30-s + 1.57·31-s + 1.43·32-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

\(L(1)\) \(\approx\) \(0.6376287831\)
\(L(\frac12)\) \(\approx\) \(0.6376287831\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.06T + 2T^{2} \)
5 \( 1 + 0.608T + 5T^{2} \)
11 \( 1 + 1.10T + 11T^{2} \)
13 \( 1 - 1.84T + 13T^{2} \)
17 \( 1 - 5.88T + 17T^{2} \)
19 \( 1 - 1.08T + 19T^{2} \)
29 \( 1 + 0.804T + 29T^{2} \)
31 \( 1 - 8.77T + 31T^{2} \)
37 \( 1 - 9.11T + 37T^{2} \)
41 \( 1 + 9.79T + 41T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 + 5.90T + 47T^{2} \)
53 \( 1 - 4.51T + 53T^{2} \)
59 \( 1 + 7.40T + 59T^{2} \)
61 \( 1 - 5.02T + 61T^{2} \)
67 \( 1 - 2.06T + 67T^{2} \)
71 \( 1 - 4.30T + 71T^{2} \)
73 \( 1 + 1.15T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + 7.92T + 89T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−8.432498384073886882653970271272, −8.024019338753971344339799622212, −7.39681688782931761220270723138, −6.56180757724017849512218296588, −5.79071180708948986554678472776, −4.89095606813865919867062982248, −3.90755222533992550805640249743, −2.77984710864393777247253622971, −1.51077818719431514595773456197, −0.65666206084317941630737210210, 0.65666206084317941630737210210, 1.51077818719431514595773456197, 2.77984710864393777247253622971, 3.90755222533992550805640249743, 4.89095606813865919867062982248, 5.79071180708948986554678472776, 6.56180757724017849512218296588, 7.39681688782931761220270723138, 8.024019338753971344339799622212, 8.432498384073886882653970271272

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