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  − 1.41·2-s + 1.41·3-s + 1.00·4-s − 5-s − 2.00·6-s + 1.00·9-s + 1.41·10-s + 1.41·12-s − 1.41·13-s − 1.41·15-s − 0.999·16-s − 1.41·18-s − 19-s − 1.00·20-s + 25-s + 2.00·26-s + 2.00·30-s + 1.41·32-s + 1.00·36-s + 1.41·37-s + 1.41·38-s − 2.00·39-s − 1.00·45-s − 1.41·48-s + 49-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.41·3-s + 1.00·4-s − 5-s − 2.00·6-s + 1.00·9-s + 1.41·10-s + 1.41·12-s − 1.41·13-s − 1.41·15-s − 0.999·16-s − 1.41·18-s − 19-s − 1.00·20-s + 25-s + 2.00·26-s + 2.00·30-s + 1.41·32-s + 1.00·36-s + 1.41·37-s + 1.41·38-s − 2.00·39-s − 1.00·45-s − 1.41·48-s + 49-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  =  \((2,\ 95,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.3812190448\)
\(L(\frac12)\)  \(\approx\)  \(0.3812190448\)
\(L(1)\)   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 \( 1 + 1.41T + T^{2} \)
3 \( 1 - 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−14.61957954625341714164664019560, −13.27042701489436960383904648520, −11.97440438418630739376367126084, −10.64837931859429375334750285042, −9.566156585394324383794480118050, −8.691956235481506596317612803994, −7.88660575820453984407934978439, −7.14472277853861817875692871455, −4.29396961792893118367281974002, −2.51518215952811900062369427304, 2.51518215952811900062369427304, 4.29396961792893118367281974002, 7.14472277853861817875692871455, 7.88660575820453984407934978439, 8.691956235481506596317612803994, 9.566156585394324383794480118050, 10.64837931859429375334750285042, 11.97440438418630739376367126084, 13.27042701489436960383904648520, 14.61957954625341714164664019560

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