Properties

Label 2-3332-1.1-c1-0-17
Degree $2$
Conductor $3332$
Sign $1$
Analytic cond. $26.6061$
Root an. cond. $5.15811$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4·5-s − 2·9-s − 3·11-s + 5·13-s − 4·15-s + 17-s − 6·19-s − 4·23-s + 11·25-s + 5·27-s + 8·29-s + 3·33-s − 8·37-s − 5·39-s + 8·41-s + 10·43-s − 8·45-s + 2·47-s − 51-s + 3·53-s − 12·55-s + 6·57-s − 2·59-s + 8·61-s + 20·65-s + 14·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s − 2/3·9-s − 0.904·11-s + 1.38·13-s − 1.03·15-s + 0.242·17-s − 1.37·19-s − 0.834·23-s + 11/5·25-s + 0.962·27-s + 1.48·29-s + 0.522·33-s − 1.31·37-s − 0.800·39-s + 1.24·41-s + 1.52·43-s − 1.19·45-s + 0.291·47-s − 0.140·51-s + 0.412·53-s − 1.61·55-s + 0.794·57-s − 0.260·59-s + 1.02·61-s + 2.48·65-s + 1.71·67-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: yes
Arithmetic: yes
Character: $\chi_{3332} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.004493628\)
\(L(\frac12)\) \(\approx\) \(2.004493628\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.592745901478006067413909081133, −8.152865808465919495805947046937, −6.71433346291073231637249834347, −6.26639307836174474805434261592, −5.65155783123102046970331334828, −5.17372327145834137805046453831, −4.02612867331051694119474885614, −2.74021814560234606035472977391, −2.10933951911823627095213212569, −0.877247665806075906579844897195, 0.877247665806075906579844897195, 2.10933951911823627095213212569, 2.74021814560234606035472977391, 4.02612867331051694119474885614, 5.17372327145834137805046453831, 5.65155783123102046970331334828, 6.26639307836174474805434261592, 6.71433346291073231637249834347, 8.152865808465919495805947046937, 8.592745901478006067413909081133

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