Properties

Label 2-3332-68.67-c0-0-3
Degree $2$
Conductor $3332$
Sign $1$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 9-s + 2·13-s + 16-s − 17-s + 18-s + 25-s − 2·26-s − 32-s + 34-s − 36-s − 50-s + 2·52-s + 2·53-s + 64-s − 68-s + 72-s + 81-s + 2·89-s + 100-s + 2·101-s − 2·104-s − 2·106-s − 2·117-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s − 9-s + 2·13-s + 16-s − 17-s + 18-s + 25-s − 2·26-s − 32-s + 34-s − 36-s − 50-s + 2·52-s + 2·53-s + 64-s − 68-s + 72-s + 81-s + 2·89-s + 100-s + 2·101-s − 2·104-s − 2·106-s − 2·117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3332} (883, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7942650089\)
\(L(\frac12)\) \(\approx\) \(0.7942650089\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 \)
17 \( 1 + T \)
good3 \( 1 + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
13 \( ( 1 - T )^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−8.734033661653060851565056647550, −8.398598074443980036737263971839, −7.45073921304838053001412056658, −6.51371192282355604081864635128, −6.12020281721816720237542000126, −5.22844880036876663089043014736, −3.90737991840690673651127826714, −3.08420048273440997718033667649, −2.12851263876107765259720395327, −0.922737046567251908763524139817, 0.922737046567251908763524139817, 2.12851263876107765259720395327, 3.08420048273440997718033667649, 3.90737991840690673651127826714, 5.22844880036876663089043014736, 6.12020281721816720237542000126, 6.51371192282355604081864635128, 7.45073921304838053001412056658, 8.398598074443980036737263971839, 8.734033661653060851565056647550

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