Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4.70·2-s + 3·3-s + 14.1·4-s − 5·5-s + 14.1·6-s + 7·7-s + 28.7·8-s + 9·9-s − 23.5·10-s + 24.5·11-s + 42.3·12-s − 35.0·13-s + 32.9·14-s − 15·15-s + 22.1·16-s − 18.4·17-s + 42.3·18-s − 67.4·19-s − 70.5·20-s + 21·21-s + 115.·22-s − 145.·23-s + 86.1·24-s + 25·25-s − 164.·26-s + 27·27-s + 98.7·28-s + ⋯
L(s)  = 1  + 1.66·2-s + 0.577·3-s + 1.76·4-s − 0.447·5-s + 0.959·6-s + 0.377·7-s + 1.26·8-s + 0.333·9-s − 0.743·10-s + 0.674·11-s + 1.01·12-s − 0.747·13-s + 0.628·14-s − 0.258·15-s + 0.345·16-s − 0.262·17-s + 0.554·18-s − 0.813·19-s − 0.788·20-s + 0.218·21-s + 1.12·22-s − 1.32·23-s + 0.732·24-s + 0.200·25-s − 1.24·26-s + 0.192·27-s + 0.666·28-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  =  \((2,\ 105,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(4.18876\)
\(L(\frac12)\)  \(\approx\)  \(4.18876\)
\(L(\frac{5}{2})\)   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 \( 1 - 4.70T + 8T^{2} \)
11 \( 1 - 24.5T + 1.33e3T^{2} \)
13 \( 1 + 35.0T + 2.19e3T^{2} \)
17 \( 1 + 18.4T + 4.91e3T^{2} \)
19 \( 1 + 67.4T + 6.85e3T^{2} \)
23 \( 1 + 145.T + 1.21e4T^{2} \)
29 \( 1 - 214.T + 2.43e4T^{2} \)
31 \( 1 + 88.6T + 2.97e4T^{2} \)
37 \( 1 - 162.T + 5.06e4T^{2} \)
41 \( 1 + 337.T + 6.89e4T^{2} \)
43 \( 1 - 122.T + 7.95e4T^{2} \)
47 \( 1 - 354.T + 1.03e5T^{2} \)
53 \( 1 - 676.T + 1.48e5T^{2} \)
59 \( 1 - 501.T + 2.05e5T^{2} \)
61 \( 1 + 708.T + 2.26e5T^{2} \)
67 \( 1 + 907.T + 3.00e5T^{2} \)
71 \( 1 - 430.T + 3.57e5T^{2} \)
73 \( 1 - 41.3T + 3.89e5T^{2} \)
79 \( 1 - 890.T + 4.93e5T^{2} \)
83 \( 1 + 1.05e3T + 5.71e5T^{2} \)
89 \( 1 - 1.47e3T + 7.04e5T^{2} \)
97 \( 1 - 555.T + 9.12e5T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.45661982419435445974030247562, −12.31294012545683469886175642022, −11.74340904356697436312554036667, −10.38833133752295853137547898109, −8.762382598156180586820762710996, −7.41717170158120664380132296828, −6.24001980998692394717884292756, −4.71333953154907923884078271650, −3.82511888816381233528132119600, −2.32086255885988398607874075783, 2.32086255885988398607874075783, 3.82511888816381233528132119600, 4.71333953154907923884078271650, 6.24001980998692394717884292756, 7.41717170158120664380132296828, 8.762382598156180586820762710996, 10.38833133752295853137547898109, 11.74340904356697436312554036667, 12.31294012545683469886175642022, 13.45661982419435445974030247562

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