Properties

Label 2-2736-1.1-c1-0-29
Degree $2$
Conductor $2736$
Sign $-1$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 7-s + 3·11-s − 4·13-s + 3·17-s − 19-s + 4·25-s − 6·29-s + 4·31-s − 3·35-s + 2·37-s + 6·41-s + 43-s − 3·47-s − 6·49-s − 12·53-s − 9·55-s − 6·59-s − 61-s + 12·65-s + 4·67-s + 6·71-s − 7·73-s + 3·77-s − 8·79-s + 12·83-s − 9·85-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.377·7-s + 0.904·11-s − 1.10·13-s + 0.727·17-s − 0.229·19-s + 4/5·25-s − 1.11·29-s + 0.718·31-s − 0.507·35-s + 0.328·37-s + 0.937·41-s + 0.152·43-s − 0.437·47-s − 6/7·49-s − 1.64·53-s − 1.21·55-s − 0.781·59-s − 0.128·61-s + 1.48·65-s + 0.488·67-s + 0.712·71-s − 0.819·73-s + 0.341·77-s − 0.900·79-s + 1.31·83-s − 0.976·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2736} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−8.169758757825737858267666465656, −7.80441727232688730491090849584, −7.09896358802950320449173631503, −6.24695455384273016222733694355, −5.16165536636701751832135979028, −4.38988833936187387176065759569, −3.73798930153556479954881225054, −2.76577588204614961513690244188, −1.40889938157129298285707376171, 0, 1.40889938157129298285707376171, 2.76577588204614961513690244188, 3.73798930153556479954881225054, 4.38988833936187387176065759569, 5.16165536636701751832135979028, 6.24695455384273016222733694355, 7.09896358802950320449173631503, 7.80441727232688730491090849584, 8.169758757825737858267666465656

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