Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.61·2-s + 0.618·4-s − 5-s − 7-s − 2.23·8-s − 1.61·10-s − 4.23·11-s + 1.38·13-s − 1.61·14-s − 4.85·16-s − 1.61·17-s − 7.09·19-s − 0.618·20-s − 6.85·22-s − 5.38·23-s + 25-s + 2.23·26-s − 0.618·28-s + 9.56·29-s + 6.70·31-s − 3.38·32-s − 2.61·34-s + 35-s + 6.70·37-s − 11.4·38-s + 2.23·40-s − 8.09·41-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.309·4-s − 0.447·5-s − 0.377·7-s − 0.790·8-s − 0.511·10-s − 1.27·11-s + 0.383·13-s − 0.432·14-s − 1.21·16-s − 0.392·17-s − 1.62·19-s − 0.138·20-s − 1.46·22-s − 1.12·23-s + 0.200·25-s + 0.438·26-s − 0.116·28-s + 1.77·29-s + 1.20·31-s − 0.597·32-s − 0.448·34-s + 0.169·35-s + 1.10·37-s − 1.86·38-s + 0.353·40-s − 1.26·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{945} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 945,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
good2 \( 1 - 1.61T + 2T^{2} \)
11 \( 1 + 4.23T + 11T^{2} \)
13 \( 1 - 1.38T + 13T^{2} \)
17 \( 1 + 1.61T + 17T^{2} \)
19 \( 1 + 7.09T + 19T^{2} \)
23 \( 1 + 5.38T + 23T^{2} \)
29 \( 1 - 9.56T + 29T^{2} \)
31 \( 1 - 6.70T + 31T^{2} \)
37 \( 1 - 6.70T + 37T^{2} \)
41 \( 1 + 8.09T + 41T^{2} \)
43 \( 1 + 9.94T + 43T^{2} \)
47 \( 1 + 11T + 47T^{2} \)
53 \( 1 - 4.38T + 53T^{2} \)
59 \( 1 + 1.29T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 3.85T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 - 5.61T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + 1.47T + 89T^{2} \)
97 \( 1 + 3.32T + 97T^{2} \)
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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

−9.834814892749257393832066142655, −8.455068163802194315339698757933, −8.177582226516844944927988643487, −6.62652671638674137004661111524, −6.19647478656595924400803890664, −4.96678938605535477894573788617, −4.38999148689490299031447079082, −3.33569908990296030142128982539, −2.40997672140956335910460485092, 0, 2.40997672140956335910460485092, 3.33569908990296030142128982539, 4.38999148689490299031447079082, 4.96678938605535477894573788617, 6.19647478656595924400803890664, 6.62652671638674137004661111524, 8.177582226516844944927988643487, 8.455068163802194315339698757933, 9.834814892749257393832066142655

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