Properties

Degree $2$
Conductor $235200$
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 + 9-s − 5·11-s + 2·13-s − 2·17-s + 6·19-s − 23-s − 27-s − 3·29-s − 4·31-s + 5·33-s − 5·37-s − 2·39-s − 4·41-s − 7·43-s − 10·47-s + 2·51-s + 2·53-s − 6·57-s + 10·59-s + 8·61-s + 7·67-s + 69-s − 3·71-s + 2·73-s − 11·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.50·11-s + 0.554·13-s − 0.485·17-s + 1.37·19-s − 0.208·23-s − 0.192·27-s − 0.557·29-s − 0.718·31-s + 0.870·33-s − 0.821·37-s − 0.320·39-s − 0.624·41-s − 1.06·43-s − 1.45·47-s + 0.280·51-s + 0.274·53-s − 0.794·57-s + 1.30·59-s + 1.02·61-s + 0.855·67-s + 0.120·69-s − 0.356·71-s + 0.234·73-s − 1.23·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(235200\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{235200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−13.05420708096866, −12.82970842385163, −12.20458988234109, −11.56717029923621, −11.33575041420932, −10.96121922038579, −10.26976622061513, −9.945219740337482, −9.670748983368295, −8.777082582128609, −8.465834139792659, −7.984301268538136, −7.336955968713031, −7.038871198865649, −6.528335540648669, −5.690180527944584, −5.543543224272342, −5.032388152580366, −4.570223044062125, −3.737468771692829, −3.365844906400656, −2.752403808424511, −1.995670870963202, −1.529033253592777, −0.6340867099835869, 0, 0.6340867099835869, 1.529033253592777, 1.995670870963202, 2.752403808424511, 3.365844906400656, 3.737468771692829, 4.570223044062125, 5.032388152580366, 5.543543224272342, 5.690180527944584, 6.528335540648669, 7.038871198865649, 7.336955968713031, 7.984301268538136, 8.465834139792659, 8.777082582128609, 9.670748983368295, 9.945219740337482, 10.26976622061513, 10.96121922038579, 11.33575041420932, 11.56717029923621, 12.20458988234109, 12.82970842385163, 13.05420708096866

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