Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.82·2-s − 3·3-s − 4.65·4-s + 5·5-s − 5.48·6-s − 7·7-s − 23.1·8-s + 9·9-s + 9.14·10-s − 64.5·11-s + 13.9·12-s − 32.3·13-s − 12.7·14-s − 15·15-s − 5.05·16-s − 56.3·17-s + 16.4·18-s − 2.74·19-s − 23.2·20-s + 21·21-s − 118.·22-s + 88.1·23-s + 69.4·24-s + 25·25-s − 59.1·26-s − 27·27-s + 32.5·28-s + ⋯
L(s)  = 1  + 0.646·2-s − 0.577·3-s − 0.582·4-s + 0.447·5-s − 0.373·6-s − 0.377·7-s − 1.02·8-s + 0.333·9-s + 0.289·10-s − 1.76·11-s + 0.336·12-s − 0.690·13-s − 0.244·14-s − 0.258·15-s − 0.0790·16-s − 0.803·17-s + 0.215·18-s − 0.0331·19-s − 0.260·20-s + 0.218·21-s − 1.14·22-s + 0.799·23-s + 0.590·24-s + 0.200·25-s − 0.446·26-s − 0.192·27-s + 0.220·28-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  =  \(1\)
Selberg data  =  \((2,\ 105,\ (\ :3/2),\ -1)\)
\(L(2)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{5}{2})\)   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 \( 1 - 1.82T + 8T^{2} \)
11 \( 1 + 64.5T + 1.33e3T^{2} \)
13 \( 1 + 32.3T + 2.19e3T^{2} \)
17 \( 1 + 56.3T + 4.91e3T^{2} \)
19 \( 1 + 2.74T + 6.85e3T^{2} \)
23 \( 1 - 88.1T + 1.21e4T^{2} \)
29 \( 1 - 246.T + 2.43e4T^{2} \)
31 \( 1 + 110.T + 2.97e4T^{2} \)
37 \( 1 - 120.T + 5.06e4T^{2} \)
41 \( 1 + 176.T + 6.89e4T^{2} \)
43 \( 1 + 443.T + 7.95e4T^{2} \)
47 \( 1 + 345.T + 1.03e5T^{2} \)
53 \( 1 - 260.T + 1.48e5T^{2} \)
59 \( 1 - 628.T + 2.05e5T^{2} \)
61 \( 1 + 115.T + 2.26e5T^{2} \)
67 \( 1 + 951.T + 3.00e5T^{2} \)
71 \( 1 - 356.T + 3.57e5T^{2} \)
73 \( 1 + 656.T + 3.89e5T^{2} \)
79 \( 1 - 440.T + 4.93e5T^{2} \)
83 \( 1 + 54.4T + 5.71e5T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + 724.T + 9.12e5T^{2} \)
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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

−13.08292458916580721099457241307, −11.98572694454539870279324252193, −10.58069027880479118600035779748, −9.718913732199968094925560978861, −8.382732867907255423730636167999, −6.78724653057891845595441544403, −5.46566543781079009701801300000, −4.70989698121031038534607877438, −2.82999919904791572750541919205, 0, 2.82999919904791572750541919205, 4.70989698121031038534607877438, 5.46566543781079009701801300000, 6.78724653057891845595441544403, 8.382732867907255423730636167999, 9.718913732199968094925560978861, 10.58069027880479118600035779748, 11.98572694454539870279324252193, 13.08292458916580721099457241307

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