Properties

Degree 2
Conductor $ 5 \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  − 4-s + 5-s − 9-s − 2·11-s + 16-s + 19-s − 20-s + 25-s + 36-s + 2·44-s − 45-s + 49-s − 2·55-s − 2·61-s − 64-s − 76-s + 80-s + 81-s + 95-s + 2·99-s − 100-s − 2·101-s + ⋯
L(s)  = 1  − 4-s + 5-s − 9-s − 2·11-s + 16-s + 19-s − 20-s + 25-s + 36-s + 2·44-s − 45-s + 49-s − 2·55-s − 2·61-s − 64-s − 76-s + 80-s + 81-s + 95-s + 2·99-s − 100-s − 2·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(95\)    =    \(5 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{95} (94, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 95,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.4985172696\)
\(L(\frac12)\)  \(\approx\)  \(0.4985172696\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;19\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{5,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 + T^{2} \)
3 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 + T )^{2} \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T^{2} \)
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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

−13.90530768021683048703931500792, −13.47265417168699841911242379081, −12.42493324594992440640465564801, −10.80601861381863672990663374102, −9.891081373460128606623069373424, −8.844029451101821452107687940364, −7.76012765378292796390012623669, −5.76418806130242862195870775024, −5.05764801474219645507661583018, −2.87350641693744207631912289409, 2.87350641693744207631912289409, 5.05764801474219645507661583018, 5.76418806130242862195870775024, 7.76012765378292796390012623669, 8.844029451101821452107687940364, 9.891081373460128606623069373424, 10.80601861381863672990663374102, 12.42493324594992440640465564801, 13.47265417168699841911242379081, 13.90530768021683048703931500792

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