Properties

Label 2-3332-1.1-c1-0-16
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  − 3.08·3-s − 3.25·5-s + 6.53·9-s − 5.12·11-s − 5.13·13-s + 10.0·15-s + 17-s + 5.25·19-s + 6.97·23-s + 5.59·25-s − 10.9·27-s − 3.39·29-s − 2.35·31-s + 15.8·33-s + 2.60·37-s + 15.8·39-s + 5.70·41-s + 1.18·43-s − 21.2·45-s − 4.70·47-s − 3.08·51-s + 10.0·53-s + 16.6·55-s − 16.2·57-s + 13.0·59-s − 1.78·61-s + 16.7·65-s + ⋯
L(s)  = 1  − 1.78·3-s − 1.45·5-s + 2.17·9-s − 1.54·11-s − 1.42·13-s + 2.59·15-s + 0.242·17-s + 1.20·19-s + 1.45·23-s + 1.11·25-s − 2.10·27-s − 0.630·29-s − 0.423·31-s + 2.75·33-s + 0.428·37-s + 2.53·39-s + 0.891·41-s + 0.180·43-s − 3.17·45-s − 0.685·47-s − 0.432·51-s + 1.37·53-s + 2.24·55-s − 2.14·57-s + 1.70·59-s − 0.228·61-s + 2.07·65-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 \( 1 + 3.08T + 3T^{2} \)
5 \( 1 + 3.25T + 5T^{2} \)
11 \( 1 + 5.12T + 11T^{2} \)
13 \( 1 + 5.13T + 13T^{2} \)
19 \( 1 - 5.25T + 19T^{2} \)
23 \( 1 - 6.97T + 23T^{2} \)
29 \( 1 + 3.39T + 29T^{2} \)
31 \( 1 + 2.35T + 31T^{2} \)
37 \( 1 - 2.60T + 37T^{2} \)
41 \( 1 - 5.70T + 41T^{2} \)
43 \( 1 - 1.18T + 43T^{2} \)
47 \( 1 + 4.70T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 + 1.78T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 - 9.21T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 - 2.05T + 79T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 - 6.91T + 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−7.74244255936911754658497742022, −7.49751357037577562621814826886, −6.95948834614030890712500176458, −5.76479789581663457510875749905, −5.07650205165408482468513648448, −4.78459252506694134239086194339, −3.70113683046593495574441751122, −2.61609548547692025455414834909, −0.855518317264551150864599140187, 0, 0.855518317264551150864599140187, 2.61609548547692025455414834909, 3.70113683046593495574441751122, 4.78459252506694134239086194339, 5.07650205165408482468513648448, 5.76479789581663457510875749905, 6.95948834614030890712500176458, 7.49751357037577562621814826886, 7.74244255936911754658497742022

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