Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.70·2-s + 3·3-s − 5.10·4-s − 5·5-s − 5.10·6-s + 7·7-s + 22.2·8-s + 9·9-s + 8.50·10-s + 37.4·11-s − 15.3·12-s + 29.0·13-s − 11.9·14-s − 15·15-s + 2.89·16-s + 58.4·17-s − 15.3·18-s − 54.5·19-s + 25.5·20-s + 21·21-s − 63.6·22-s + 161.·23-s + 66.8·24-s + 25·25-s − 49.3·26-s + 27·27-s − 35.7·28-s + ⋯
L(s)  = 1  − 0.601·2-s + 0.577·3-s − 0.638·4-s − 0.447·5-s − 0.347·6-s + 0.377·7-s + 0.985·8-s + 0.333·9-s + 0.269·10-s + 1.02·11-s − 0.368·12-s + 0.619·13-s − 0.227·14-s − 0.258·15-s + 0.0452·16-s + 0.833·17-s − 0.200·18-s − 0.659·19-s + 0.285·20-s + 0.218·21-s − 0.616·22-s + 1.46·23-s + 0.568·24-s + 0.200·25-s − 0.372·26-s + 0.192·27-s − 0.241·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{105} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(1.25861\)
\(L(\frac12)\)  \(\approx\)  \(1.25861\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - 3T \)
5 \( 1 + 5T \)
7 \( 1 - 7T \)
good2 \( 1 + 1.70T + 8T^{2} \)
11 \( 1 - 37.4T + 1.33e3T^{2} \)
13 \( 1 - 29.0T + 2.19e3T^{2} \)
17 \( 1 - 58.4T + 4.91e3T^{2} \)
19 \( 1 + 54.5T + 6.85e3T^{2} \)
23 \( 1 - 161.T + 1.21e4T^{2} \)
29 \( 1 - 137.T + 2.43e4T^{2} \)
31 \( 1 - 154.T + 2.97e4T^{2} \)
37 \( 1 + 350.T + 5.06e4T^{2} \)
41 \( 1 - 353.T + 6.89e4T^{2} \)
43 \( 1 + 518.T + 7.95e4T^{2} \)
47 \( 1 + 542.T + 1.03e5T^{2} \)
53 \( 1 - 305.T + 1.48e5T^{2} \)
59 \( 1 - 14.6T + 2.05e5T^{2} \)
61 \( 1 + 171.T + 2.26e5T^{2} \)
67 \( 1 - 551.T + 3.00e5T^{2} \)
71 \( 1 + 120.T + 3.57e5T^{2} \)
73 \( 1 - 284.T + 3.89e5T^{2} \)
79 \( 1 - 941.T + 4.93e5T^{2} \)
83 \( 1 - 377.T + 5.71e5T^{2} \)
89 \( 1 + 677.T + 7.04e5T^{2} \)
97 \( 1 + 1.22e3T + 9.12e5T^{2} \)
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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

−13.42975125679078246543569448949, −12.25656411438848290036343603269, −10.99283288889158405781600786128, −9.849320605974813936721103228053, −8.760606868012696065345540490324, −8.165518684722344472648947709944, −6.81635547436989365399984430237, −4.83096030102797826311073591617, −3.57515772035303027225893561748, −1.20229461864017516428374702022, 1.20229461864017516428374702022, 3.57515772035303027225893561748, 4.83096030102797826311073591617, 6.81635547436989365399984430237, 8.165518684722344472648947709944, 8.760606868012696065345540490324, 9.849320605974813936721103228053, 10.99283288889158405781600786128, 12.25656411438848290036343603269, 13.42975125679078246543569448949

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