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 + 6.40·13-s − 2.58·14-s − 15-s + 8.70·16-s − 3.70·17-s + 2.58·18-s + 7.24·19-s − 4.70·20-s − 21-s − 2.58·22-s − 7.33·23-s + 6.99·24-s + 25-s + 16.5·26-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 + 1.77·13-s − 0.691·14-s − 0.258·15-s + 2.17·16-s − 0.897·17-s + 0.610·18-s + 1.66·19-s − 1.05·20-s − 0.218·21-s − 0.551·22-s − 1.52·23-s + 1.42·24-s + 0.200·25-s + 3.25·26-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  =  \((2,\ 1155,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(5.536754837\)
\(L(\frac12)\)  \(\approx\)  \(5.536754837\)
\(L(\frac{3}{2})\)   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 \( 1 - 6.40T + 13T^{2} \)
17 \( 1 + 3.70T + 17T^{2} \)
19 \( 1 - 7.24T + 19T^{2} \)
23 \( 1 + 7.33T + 23T^{2} \)
29 \( 1 + 8.10T + 29T^{2} \)
31 \( 1 - 6.77T + 31T^{2} \)
37 \( 1 + 3.95T + 37T^{2} \)
41 \( 1 + 2.31T + 41T^{2} \)
43 \( 1 + 7.65T + 43T^{2} \)
47 \( 1 - 1.45T + 47T^{2} \)
53 \( 1 - 9.10T + 53T^{2} \)
59 \( 1 + 2.92T + 59T^{2} \)
61 \( 1 + 8.42T + 61T^{2} \)
67 \( 1 + 8.72T + 67T^{2} \)
71 \( 1 + 4.40T + 71T^{2} \)
73 \( 1 - 3.40T + 73T^{2} \)
79 \( 1 - 4.40T + 79T^{2} \)
83 \( 1 + 8.96T + 83T^{2} \)
89 \( 1 - 0.206T + 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−10.01550126147392317635638893160, −8.827163855960185217621945755359, −7.909358773755266253552793924634, −7.07230571138125899395847677134, −6.21146211996191133827601478262, −5.49785674740919231189777543041, −4.35541577843346628666344730143, −3.65477393895859259061618253869, −3.02548898860575796164904232976, −1.73395395791610990103311187312, 1.73395395791610990103311187312, 3.02548898860575796164904232976, 3.65477393895859259061618253869, 4.35541577843346628666344730143, 5.49785674740919231189777543041, 6.21146211996191133827601478262, 7.07230571138125899395847677134, 7.909358773755266253552793924634, 8.827163855960185217621945755359, 10.01550126147392317635638893160

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