Properties

Degree $2$
Conductor $1734$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s + 9-s + 2·10-s + 4·11-s − 12-s − 2·13-s − 2·15-s + 16-s + 18-s + 4·19-s + 2·20-s + 4·22-s − 24-s − 25-s − 2·26-s − 27-s + 10·29-s − 2·30-s − 8·31-s + 32-s − 4·33-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s − 0.516·15-s + 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.852·22-s − 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 1.85·29-s − 0.365·30-s − 1.43·31-s + 0.176·32-s − 0.696·33-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1734\)    =    \(2 \cdot 3 \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1734} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1734,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.870996888\)
\(L(\frac12)\) \(\approx\) \(2.870996888\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
17 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−9.547040289033150690294822441027, −8.613628586363707276957269542969, −7.37401646943555492580120617253, −6.73915837939670470281220727226, −5.97447010434826597680072551041, −5.32734843201609780311424178486, −4.47367404653766349933628761316, −3.49143456848613430686623125279, −2.28380910146500603557363711542, −1.19155942635042385588089355891, 1.19155942635042385588089355891, 2.28380910146500603557363711542, 3.49143456848613430686623125279, 4.47367404653766349933628761316, 5.32734843201609780311424178486, 5.97447010434826597680072551041, 6.73915837939670470281220727226, 7.37401646943555492580120617253, 8.613628586363707276957269542969, 9.547040289033150690294822441027

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