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  + 0.236·2-s + 3·3-s − 7.94·4-s − 5·5-s + 0.708·6-s − 7·7-s − 3.76·8-s + 9·9-s − 1.18·10-s − 50.4·11-s − 23.8·12-s − 80.9·13-s − 1.65·14-s − 15·15-s + 62.6·16-s + 76.3·17-s + 2.12·18-s + 4.13·19-s + 39.7·20-s − 21·21-s − 11.9·22-s − 204.·23-s − 11.2·24-s + 25·25-s − 19.1·26-s + 27·27-s + 55.6·28-s + ⋯
L(s)  = 1  + 0.0834·2-s + 0.577·3-s − 0.993·4-s − 0.447·5-s + 0.0481·6-s − 0.377·7-s − 0.166·8-s + 0.333·9-s − 0.0373·10-s − 1.38·11-s − 0.573·12-s − 1.72·13-s − 0.0315·14-s − 0.258·15-s + 0.979·16-s + 1.08·17-s + 0.0278·18-s + 0.0499·19-s + 0.444·20-s − 0.218·21-s − 0.115·22-s − 1.85·23-s − 0.0960·24-s + 0.200·25-s − 0.144·26-s + 0.192·27-s + 0.375·28-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 \( 1 - 0.236T + 8T^{2} \)
11 \( 1 + 50.4T + 1.33e3T^{2} \)
13 \( 1 + 80.9T + 2.19e3T^{2} \)
17 \( 1 - 76.3T + 4.91e3T^{2} \)
19 \( 1 - 4.13T + 6.85e3T^{2} \)
23 \( 1 + 204.T + 1.21e4T^{2} \)
29 \( 1 + 91.1T + 2.43e4T^{2} \)
31 \( 1 - 198.T + 2.97e4T^{2} \)
37 \( 1 - 155.T + 5.06e4T^{2} \)
41 \( 1 + 156.T + 6.89e4T^{2} \)
43 \( 1 - 354.T + 7.95e4T^{2} \)
47 \( 1 + 175.T + 1.03e5T^{2} \)
53 \( 1 - 200.T + 1.48e5T^{2} \)
59 \( 1 + 312.T + 2.05e5T^{2} \)
61 \( 1 + 154.T + 2.26e5T^{2} \)
67 \( 1 - 734.T + 3.00e5T^{2} \)
71 \( 1 + 678.T + 3.57e5T^{2} \)
73 \( 1 + 60.8T + 3.89e5T^{2} \)
79 \( 1 + 1.28e3T + 4.93e5T^{2} \)
83 \( 1 - 116.T + 5.71e5T^{2} \)
89 \( 1 + 916.T + 7.04e5T^{2} \)
97 \( 1 + 1.41e3T + 9.12e5T^{2} \)
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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

−12.78558300354075809399407693352, −12.09629816446006541908408907175, −10.16767996106242022361660541788, −9.684910701443008864898681857926, −8.182929901094436683442313395985, −7.56005056030982890340253285064, −5.51723347212477855927934184572, −4.29591832719651791160000479572, −2.81449756126596726250098903133, 0, 2.81449756126596726250098903133, 4.29591832719651791160000479572, 5.51723347212477855927934184572, 7.56005056030982890340253285064, 8.182929901094436683442313395985, 9.684910701443008864898681857926, 10.16767996106242022361660541788, 12.09629816446006541908408907175, 12.78558300354075809399407693352

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