Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
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-s + 3-s − 4-s + 5-s + 6-s + 7-s − 3·8-s + 9-s + 10-s − 12-s − 6·13-s + 14-s + 15-s − 16-s + 2·17-s + 18-s − 8·19-s − 20-s + 21-s + 8·23-s − 3·24-s + 25-s − 6·26-s + 27-s − 28-s − 2·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 1.66·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.218·21-s + 1.66·23-s − 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{105} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.465863363$
$L(\frac12)$  $\approx$  $1.465863363$
$L(\frac{3}{2})$   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 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−19.36715797509328, −18.77735506590928, −17.46635077807532, −16.94525385553172, −15.00236141561966, −14.85464607432052, −13.78714664919276, −12.89916852888262, −12.11417775373725, −10.47975989489334, −9.386500711260632, −8.447489066361172, −6.979443823405172, −5.375733425527364, −4.333090589487554, −2.632947564799944, 2.632947564799944, 4.333090589487554, 5.375733425527364, 6.979443823405172, 8.447489066361172, 9.386500711260632, 10.47975989489334, 12.11417775373725, 12.89916852888262, 13.78714664919276, 14.85464607432052, 15.00236141561966, 16.94525385553172, 17.46635077807532, 18.77735506590928, 19.36715797509328

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