Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.58·2-s + 3-s + 4.70·4-s − 5-s − 2.58·6-s − 7-s − 6.99·8-s + 9-s + 2.58·10-s − 11-s + 4.70·12-s − 2.40·13-s + 2.58·14-s − 15-s + 8.70·16-s − 3.70·17-s − 2.58·18-s + 4.15·19-s − 4.70·20-s − 21-s + 2.58·22-s − 0.0692·23-s − 6.99·24-s + 25-s + 6.22·26-s + 27-s − 4.70·28-s + ⋯
L(s)  = 1  − 1.83·2-s + 0.577·3-s + 2.35·4-s − 0.447·5-s − 1.05·6-s − 0.377·7-s − 2.47·8-s + 0.333·9-s + 0.818·10-s − 0.301·11-s + 1.35·12-s − 0.666·13-s + 0.691·14-s − 0.258·15-s + 2.17·16-s − 0.897·17-s − 0.610·18-s + 0.953·19-s − 1.05·20-s − 0.218·21-s + 0.551·22-s − 0.0144·23-s − 1.42·24-s + 0.200·25-s + 1.22·26-s + 0.192·27-s − 0.888·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1155} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1155,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.6591865022\)
\(L(\frac12)\)  \(\approx\)  \(0.6591865022\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + T \)
good2 \( 1 + 2.58T + 2T^{2} \)
13 \( 1 + 2.40T + 13T^{2} \)
17 \( 1 + 3.70T + 17T^{2} \)
19 \( 1 - 4.15T + 19T^{2} \)
23 \( 1 + 0.0692T + 23T^{2} \)
29 \( 1 - 0.703T + 29T^{2} \)
31 \( 1 - 5.22T + 31T^{2} \)
37 \( 1 - 7.95T + 37T^{2} \)
41 \( 1 - 2.31T + 41T^{2} \)
43 \( 1 - 4.24T + 43T^{2} \)
47 \( 1 - 13.3T + 47T^{2} \)
53 \( 1 + 8.51T + 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 - 5.02T + 61T^{2} \)
67 \( 1 - 4.72T + 67T^{2} \)
71 \( 1 - 4.40T + 71T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 + 4.40T + 79T^{2} \)
83 \( 1 - 5.56T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
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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

−9.569215425675887458666642814400, −9.078817395623990327795389195183, −8.213607569143270422488818730559, −7.57436886988685560740685531690, −6.99124926173308188664323094612, −5.99669726581949430975170155912, −4.47224355619173964652126025914, −3.03622844816130731329806270061, −2.26836298564414965110307697230, −0.76246366574970015371563127252, 0.76246366574970015371563127252, 2.26836298564414965110307697230, 3.03622844816130731329806270061, 4.47224355619173964652126025914, 5.99669726581949430975170155912, 6.99124926173308188664323094612, 7.57436886988685560740685531690, 8.213607569143270422488818730559, 9.078817395623990327795389195183, 9.569215425675887458666642814400

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