Properties

Label 2-3332-1.1-c1-0-21
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  − 3·3-s + 4·5-s + 6·9-s + 11-s + 3·13-s − 12·15-s − 17-s − 2·19-s + 4·23-s + 11·25-s − 9·27-s + 8·31-s − 3·33-s + 8·37-s − 9·39-s + 10·43-s + 24·45-s − 10·47-s + 3·51-s + 3·53-s + 4·55-s + 6·57-s − 14·59-s − 8·61-s + 12·65-s − 10·67-s − 12·69-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.78·5-s + 2·9-s + 0.301·11-s + 0.832·13-s − 3.09·15-s − 0.242·17-s − 0.458·19-s + 0.834·23-s + 11/5·25-s − 1.73·27-s + 1.43·31-s − 0.522·33-s + 1.31·37-s − 1.44·39-s + 1.52·43-s + 3.57·45-s − 1.45·47-s + 0.420·51-s + 0.412·53-s + 0.539·55-s + 0.794·57-s − 1.82·59-s − 1.02·61-s + 1.48·65-s − 1.22·67-s − 1.44·69-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.663820084\)
\(L(\frac12)\) \(\approx\) \(1.663820084\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.973936402482286354207399587458, −7.69024035678898130149858331901, −6.60165596308849873907420846454, −6.22078735832051372202585427283, −5.87479884156858091839879278168, −4.93772697182014911636657086789, −4.40820492715886183282953413943, −2.87859951484990752955258598597, −1.67453831099288359567119266994, −0.914365510636910812510763279545, 0.914365510636910812510763279545, 1.67453831099288359567119266994, 2.87859951484990752955258598597, 4.40820492715886183282953413943, 4.93772697182014911636657086789, 5.87479884156858091839879278168, 6.22078735832051372202585427283, 6.60165596308849873907420846454, 7.69024035678898130149858331901, 8.973936402482286354207399587458

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