Properties

Degree 2
Conductor $ 3 \cdot 37 $
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 − 3-s + 1.00·4-s − 1.41·5-s − 1.41·6-s + 9-s − 2.00·10-s − 1.00·12-s + 1.41·15-s − 0.999·16-s + 1.41·17-s + 1.41·18-s − 1.41·20-s − 1.41·23-s + 1.00·25-s − 27-s + 1.41·29-s + 2.00·30-s − 1.41·32-s + 2.00·34-s + 1.00·36-s − 37-s − 1.41·45-s − 2.00·46-s + 0.999·48-s − 49-s + ⋯
L(s)  = 1  + 1.41·2-s − 3-s + 1.00·4-s − 1.41·5-s − 1.41·6-s + 9-s − 2.00·10-s − 1.00·12-s + 1.41·15-s − 0.999·16-s + 1.41·17-s + 1.41·18-s − 1.41·20-s − 1.41·23-s + 1.00·25-s − 27-s + 1.41·29-s + 2.00·30-s − 1.41·32-s + 2.00·34-s + 1.00·36-s − 37-s − 1.41·45-s − 2.00·46-s + 0.999·48-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(111\)    =    \(3 \cdot 37\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{111} (110, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 111,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.7286629132\)
\(L(\frac12)\)  \(\approx\)  \(0.7286629132\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;37\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;37\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
37 \( 1 + T \)
good2 \( 1 - 1.41T + T^{2} \)
5 \( 1 + 1.41T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
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

−13.86662501688781585150006845396, −12.50712601288678636476174285911, −12.08169828841517647750547962500, −11.37051472087599925796301330751, −10.09400887276507614513350913030, −8.109234277758529601641032437684, −6.88701971246819184873514860915, −5.66648124971551170411397274205, −4.53540235685744608690914638694, −3.54574592690469057076498645767, 3.54574592690469057076498645767, 4.53540235685744608690914638694, 5.66648124971551170411397274205, 6.88701971246819184873514860915, 8.109234277758529601641032437684, 10.09400887276507614513350913030, 11.37051472087599925796301330751, 12.08169828841517647750547962500, 12.50712601288678636476174285911, 13.86662501688781585150006845396

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