Properties

Label 2-3332-68.67-c0-0-6
Degree $2$
Conductor $3332$
Sign $1$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.73·3-s + 4-s + 1.73·6-s − 8-s + 1.99·9-s + 1.73·11-s − 1.73·12-s + 13-s + 16-s + 17-s − 1.99·18-s − 1.73·22-s + 1.73·24-s + 25-s − 26-s − 1.73·27-s − 32-s − 2.99·33-s − 34-s + 1.99·36-s − 1.73·39-s + 1.73·44-s − 1.73·48-s − 50-s − 1.73·51-s + 52-s + ⋯
L(s)  = 1  − 2-s − 1.73·3-s + 4-s + 1.73·6-s − 8-s + 1.99·9-s + 1.73·11-s − 1.73·12-s + 13-s + 16-s + 17-s − 1.99·18-s − 1.73·22-s + 1.73·24-s + 25-s − 26-s − 1.73·27-s − 32-s − 2.99·33-s − 34-s + 1.99·36-s − 1.73·39-s + 1.73·44-s − 1.73·48-s − 50-s − 1.73·51-s + 52-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: no
Arithmetic: yes
Character: $\chi_{3332} (883, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5741399158\)
\(L(\frac12)\) \(\approx\) \(0.5741399158\)
\(L(1)\) 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 \( 1 + 1.73T + T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 - 1.73T + T^{2} \)
13 \( 1 - T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.73T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.973293562638251319724270053544, −8.056522561671168512024461503494, −7.10350550302749743541646205977, −6.52341598717714413264105181402, −6.07660500680430133180969592635, −5.31004788033584759314288198884, −4.21815871410334199376265175633, −3.29416024314644885509815301305, −1.54204151387104326027237222343, −0.960671531228963139956870116350, 0.960671531228963139956870116350, 1.54204151387104326027237222343, 3.29416024314644885509815301305, 4.21815871410334199376265175633, 5.31004788033584759314288198884, 6.07660500680430133180969592635, 6.52341598717714413264105181402, 7.10350550302749743541646205977, 8.056522561671168512024461503494, 8.973293562638251319724270053544

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