Properties

Label 2-3332-1.1-c1-0-40
Degree $2$
Conductor $3332$
Sign $-1$
Analytic cond. $26.6061$
Root an. cond. $5.15811$
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  − 1.26·3-s + 1.77·5-s − 1.40·9-s + 0.318·11-s − 0.205·13-s − 2.24·15-s + 17-s − 5.45·19-s + 9.43·23-s − 1.85·25-s + 5.56·27-s − 8.61·29-s − 4.94·31-s − 0.402·33-s + 7.64·37-s + 0.260·39-s − 1.44·41-s − 1.98·43-s − 2.48·45-s + 1.56·47-s − 1.26·51-s − 13.5·53-s + 0.564·55-s + 6.90·57-s − 13.2·59-s − 3.37·61-s − 0.365·65-s + ⋯
L(s)  = 1  − 0.729·3-s + 0.793·5-s − 0.467·9-s + 0.0959·11-s − 0.0571·13-s − 0.579·15-s + 0.242·17-s − 1.25·19-s + 1.96·23-s − 0.370·25-s + 1.07·27-s − 1.60·29-s − 0.888·31-s − 0.0700·33-s + 1.25·37-s + 0.0416·39-s − 0.225·41-s − 0.302·43-s − 0.370·45-s + 0.228·47-s − 0.176·51-s − 1.85·53-s + 0.0761·55-s + 0.913·57-s − 1.71·59-s − 0.431·61-s − 0.0453·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(26.6061\)
Root analytic conductor: \(5.15811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3332,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
7 \( 1 \)
17 \( 1 - T \)
good3 \( 1 + 1.26T + 3T^{2} \)
5 \( 1 - 1.77T + 5T^{2} \)
11 \( 1 - 0.318T + 11T^{2} \)
13 \( 1 + 0.205T + 13T^{2} \)
19 \( 1 + 5.45T + 19T^{2} \)
23 \( 1 - 9.43T + 23T^{2} \)
29 \( 1 + 8.61T + 29T^{2} \)
31 \( 1 + 4.94T + 31T^{2} \)
37 \( 1 - 7.64T + 37T^{2} \)
41 \( 1 + 1.44T + 41T^{2} \)
43 \( 1 + 1.98T + 43T^{2} \)
47 \( 1 - 1.56T + 47T^{2} \)
53 \( 1 + 13.5T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 + 3.37T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 + 2.62T + 71T^{2} \)
73 \( 1 + 3.86T + 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + 0.00620T + 89T^{2} \)
97 \( 1 + 4.93T + 97T^{2} \)
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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

−8.295366026961139596563437139081, −7.41255745470245216651935809169, −6.54191032854964381042301539652, −5.98091344963362711951394117778, −5.32964569173687112203167316277, −4.61061291031414955249395076158, −3.46088471828285909106786329561, −2.47023775430054655132961683333, −1.43381897764040318744710026608, 0, 1.43381897764040318744710026608, 2.47023775430054655132961683333, 3.46088471828285909106786329561, 4.61061291031414955249395076158, 5.32964569173687112203167316277, 5.98091344963362711951394117778, 6.54191032854964381042301539652, 7.41255745470245216651935809169, 8.295366026961139596563437139081

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