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  − 3.82·2-s − 3·3-s + 6.65·4-s + 5·5-s + 11.4·6-s − 7·7-s + 5.14·8-s + 9·9-s − 19.1·10-s + 48.5·11-s − 19.9·12-s − 43.6·13-s + 26.7·14-s − 15·15-s − 72.9·16-s − 67.6·17-s − 34.4·18-s − 93.2·19-s + 33.2·20-s + 21·21-s − 185.·22-s − 104.·23-s − 15.4·24-s + 25·25-s + 167.·26-s − 27·27-s − 46.5·28-s + ⋯
L(s)  = 1  − 1.35·2-s − 0.577·3-s + 0.832·4-s + 0.447·5-s + 0.781·6-s − 0.377·7-s + 0.227·8-s + 0.333·9-s − 0.605·10-s + 1.33·11-s − 0.480·12-s − 0.931·13-s + 0.511·14-s − 0.258·15-s − 1.13·16-s − 0.965·17-s − 0.451·18-s − 1.12·19-s + 0.372·20-s + 0.218·21-s − 1.80·22-s − 0.944·23-s − 0.131·24-s + 0.200·25-s + 1.26·26-s − 0.192·27-s − 0.314·28-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 \( 1 + 3.82T + 8T^{2} \)
11 \( 1 - 48.5T + 1.33e3T^{2} \)
13 \( 1 + 43.6T + 2.19e3T^{2} \)
17 \( 1 + 67.6T + 4.91e3T^{2} \)
19 \( 1 + 93.2T + 6.85e3T^{2} \)
23 \( 1 + 104.T + 1.21e4T^{2} \)
29 \( 1 + 58.7T + 2.43e4T^{2} \)
31 \( 1 + 9.08T + 2.97e4T^{2} \)
37 \( 1 + 252.T + 5.06e4T^{2} \)
41 \( 1 - 276.T + 6.89e4T^{2} \)
43 \( 1 + 92.6T + 7.95e4T^{2} \)
47 \( 1 + 582.T + 1.03e5T^{2} \)
53 \( 1 - 623.T + 1.48e5T^{2} \)
59 \( 1 + 524.T + 2.05e5T^{2} \)
61 \( 1 + 352.T + 2.26e5T^{2} \)
67 \( 1 + 736.T + 3.00e5T^{2} \)
71 \( 1 + 492.T + 3.57e5T^{2} \)
73 \( 1 - 1.16e3T + 3.89e5T^{2} \)
79 \( 1 + 872.T + 4.93e5T^{2} \)
83 \( 1 + 529.T + 5.71e5T^{2} \)
89 \( 1 + 385.T + 7.04e5T^{2} \)
97 \( 1 + 463.T + 9.12e5T^{2} \)
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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

−12.52362329578303523174612218328, −11.40906351113754730786648519606, −10.35731378291167851175667760938, −9.522381603063368539181413992794, −8.655846868553477110650215124948, −7.15431105242583476581047669936, −6.25238828112403108123191162025, −4.41405209067270110225811331286, −1.85312014652769458321998380111, 0, 1.85312014652769458321998380111, 4.41405209067270110225811331286, 6.25238828112403108123191162025, 7.15431105242583476581047669936, 8.655846868553477110650215124948, 9.522381603063368539181413992794, 10.35731378291167851175667760938, 11.40906351113754730786648519606, 12.52362329578303523174612218328

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