Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 2·13-s − 15-s + 6·17-s + 8·19-s + 25-s + 27-s + 6·29-s − 4·31-s − 10·37-s − 2·39-s + 6·41-s + 4·43-s − 45-s + 6·51-s − 6·53-s + 8·57-s − 12·59-s + 10·61-s + 2·65-s + 4·67-s − 12·71-s + 10·73-s + 75-s − 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 1.45·17-s + 1.83·19-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 1.64·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 0.840·51-s − 0.824·53-s + 1.05·57-s − 1.56·59-s + 1.28·61-s + 0.248·65-s + 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.115·75-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11760\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{11760} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 11760,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.668421033\)
\(L(\frac12)\)  \(\approx\)  \(2.668421033\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−16.14828930060386, −15.93617702121665, −15.32125939879068, −14.51394661466724, −14.20288657539311, −13.80774328663217, −12.91392023252724, −12.30463122639478, −11.99242229624512, −11.30632343239036, −10.47226208397189, −9.997083055998300, −9.367965458919266, −8.854820418843128, −7.960593301944536, −7.606746052002103, −7.130108824311213, −6.225216221334624, −5.328568460355334, −4.930511681215701, −3.911281628220572, −3.289204483716040, −2.764629871397258, −1.625349451864508, −0.7591250888932235, 0.7591250888932235, 1.625349451864508, 2.764629871397258, 3.289204483716040, 3.911281628220572, 4.930511681215701, 5.328568460355334, 6.225216221334624, 7.130108824311213, 7.606746052002103, 7.960593301944536, 8.854820418843128, 9.367965458919266, 9.997083055998300, 10.47226208397189, 11.30632343239036, 11.99242229624512, 12.30463122639478, 12.91392023252724, 13.80774328663217, 14.20288657539311, 14.51394661466724, 15.32125939879068, 15.93617702121665, 16.14828930060386

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