Properties

Degree 2
Conductor $ 3 \cdot 137 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s − 5-s − 7-s + 9-s + 12-s − 15-s + 16-s − 19-s − 20-s − 21-s − 23-s + 27-s − 28-s − 29-s + 35-s + 36-s − 37-s + 2·41-s − 45-s − 47-s + 48-s + 2·53-s − 57-s − 60-s − 61-s − 63-s + ⋯
L(s)  = 1  + 3-s + 4-s − 5-s − 7-s + 9-s + 12-s − 15-s + 16-s − 19-s − 20-s − 21-s − 23-s + 27-s − 28-s − 29-s + 35-s + 36-s − 37-s + 2·41-s − 45-s − 47-s + 48-s + 2·53-s − 57-s − 60-s − 61-s − 63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 411 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 411 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(411\)    =    \(3 \cdot 137\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{411} (410, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 411,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.093491175\)
\(L(\frac12)\)  \(\approx\)  \(1.093491175\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;137\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;137\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
137 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T + T^{2} \)
53 \( ( 1 - T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )^{2} \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T + T^{2} \)
89 \( 1 + T + T^{2} \)
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

−11.47525477433014055819585806148, −10.50165252681353449018808221758, −9.670101870211429215377047795001, −8.564670901488282840563327165507, −7.70311784044715877802272763779, −7.00777087059962764495445353100, −6.01138996447162082756729418329, −4.11736566165569695423514503991, −3.34740423745780496534310739551, −2.16992743686203330013514209476, 2.16992743686203330013514209476, 3.34740423745780496534310739551, 4.11736566165569695423514503991, 6.01138996447162082756729418329, 7.00777087059962764495445353100, 7.70311784044715877802272763779, 8.564670901488282840563327165507, 9.670101870211429215377047795001, 10.50165252681353449018808221758, 11.47525477433014055819585806148

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