Properties

Degree 2
Conductor $ 2^{2} \cdot 19 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s + 9-s − 11-s − 17-s + 19-s + 2·23-s + 35-s − 43-s − 45-s − 47-s + 55-s − 61-s − 63-s − 73-s + 77-s + 81-s + 2·83-s + 85-s − 95-s − 99-s + 2·101-s − 2·115-s + 119-s + ⋯
L(s)  = 1  − 5-s − 7-s + 9-s − 11-s − 17-s + 19-s + 2·23-s + 35-s − 43-s − 45-s − 47-s + 55-s − 61-s − 63-s − 73-s + 77-s + 81-s + 2·83-s + 85-s − 95-s − 99-s + 2·101-s − 2·115-s + 119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(76\)    =    \(2^{2} \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{76} (37, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 76,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.4523541508\)
\(L(\frac12)\)  \(\approx\)  \(0.4523541508\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;19\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 - T \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T + T^{2} \)
23 \( ( 1 - T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−15.18389011950467638619571036076, −13.35974916352130316794641901241, −12.80383094195036971594468272052, −11.52365494724892801268611564374, −10.37363017330335958179980388288, −9.200011605496345198209119678485, −7.71021732429554568325384670285, −6.75966056696874593751657456611, −4.83345700374897654801370031599, −3.27623368738075539180170835921, 3.27623368738075539180170835921, 4.83345700374897654801370031599, 6.75966056696874593751657456611, 7.71021732429554568325384670285, 9.200011605496345198209119678485, 10.37363017330335958179980388288, 11.52365494724892801268611564374, 12.80383094195036971594468272052, 13.35974916352130316794641901241, 15.18389011950467638619571036076

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