Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $1$
Motivic weight 4
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 23·7-s + 191·13-s + 647·19-s + 625·25-s + 194·31-s + 2.59e3·37-s − 3.21e3·43-s − 1.87e3·49-s − 5.23e3·61-s − 8.80e3·67-s + 9.79e3·73-s − 1.23e4·79-s + 4.39e3·91-s + 9.74e3·97-s + 3.43e3·103-s + 2.20e4·109-s + ⋯
L(s)  = 1  + 0.469·7-s + 1.13·13-s + 1.79·19-s + 25-s + 0.201·31-s + 1.89·37-s − 1.73·43-s − 0.779·49-s − 1.40·61-s − 1.96·67-s + 1.83·73-s − 1.98·79-s + 0.530·91-s + 1.03·97-s + 0.323·103-s + 1.85·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  $\chi_{108} (53, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 1)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.94008\)
\(L(\frac12)\)  \(\approx\)  \(1.94008\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
7 \( 1 - 23 T + p^{4} T^{2} \)
11 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
13 \( 1 - 191 T + p^{4} T^{2} \)
17 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
19 \( 1 - 647 T + p^{4} T^{2} \)
23 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
29 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
31 \( 1 - 194 T + p^{4} T^{2} \)
37 \( 1 - 2591 T + p^{4} T^{2} \)
41 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
43 \( 1 + 3214 T + p^{4} T^{2} \)
47 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
53 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
59 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
61 \( 1 + 5233 T + p^{4} T^{2} \)
67 \( 1 + 8809 T + p^{4} T^{2} \)
71 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
73 \( 1 - 9791 T + p^{4} T^{2} \)
79 \( 1 + 12361 T + p^{4} T^{2} \)
83 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
89 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
97 \( 1 - 9743 T + p^{4} T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.10150118395261651463374501629, −11.79545170233838888637051890964, −11.01901235611302577516032265134, −9.765795152242794695298418325316, −8.603795071125446036066677679105, −7.51442878040470383663830740308, −6.12385468208258109314980868062, −4.80454257545888379074015510078, −3.21290324573003486762371726122, −1.22535312382696817150276244641, 1.22535312382696817150276244641, 3.21290324573003486762371726122, 4.80454257545888379074015510078, 6.12385468208258109314980868062, 7.51442878040470383663830740308, 8.603795071125446036066677679105, 9.765795152242794695298418325316, 11.01901235611302577516032265134, 11.79545170233838888637051890964, 13.10150118395261651463374501629

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