Properties

Degree $2$
Conductor $303600$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·7-s + 9-s − 11-s − 2·13-s − 2·19-s + 2·21-s + 23-s − 27-s − 10·29-s − 4·31-s + 33-s − 2·37-s + 2·39-s − 2·41-s + 2·43-s − 8·47-s − 3·49-s + 4·53-s + 2·57-s + 12·59-s − 6·61-s − 2·63-s + 2·67-s − 69-s + 6·73-s + 2·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.458·19-s + 0.436·21-s + 0.208·23-s − 0.192·27-s − 1.85·29-s − 0.718·31-s + 0.174·33-s − 0.328·37-s + 0.320·39-s − 0.312·41-s + 0.304·43-s − 1.16·47-s − 3/7·49-s + 0.549·53-s + 0.264·57-s + 1.56·59-s − 0.768·61-s − 0.251·63-s + 0.244·67-s − 0.120·69-s + 0.702·73-s + 0.227·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{303600} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 303600,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
11 \( 1 + T \)
23 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−12.88576829274743, −12.48290463629995, −12.06881353204020, −11.43442658787641, −11.12419753379524, −10.67861714891768, −10.14542795370608, −9.651263430232983, −9.449147442295291, −8.786001778630207, −8.309886419925173, −7.662324364522858, −7.281962064244897, −6.778251133535655, −6.420749295658727, −5.711976601379361, −5.452961985283553, −4.915884499269931, −4.319129475977780, −3.704157851169258, −3.351471198979765, −2.603341505263899, −2.033918684767319, −1.481108487144101, −0.5196748810912648, 0, 0.5196748810912648, 1.481108487144101, 2.033918684767319, 2.603341505263899, 3.351471198979765, 3.704157851169258, 4.319129475977780, 4.915884499269931, 5.452961985283553, 5.711976601379361, 6.420749295658727, 6.778251133535655, 7.281962064244897, 7.662324364522858, 8.309886419925173, 8.786001778630207, 9.449147442295291, 9.651263430232983, 10.14542795370608, 10.67861714891768, 11.12419753379524, 11.43442658787641, 12.06881353204020, 12.48290463629995, 12.88576829274743

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