Properties

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

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 9·3-s + 16·4-s − 66·5-s − 36·6-s + 176·7-s + 64·8-s + 81·9-s − 264·10-s − 60·11-s − 144·12-s − 658·13-s + 704·14-s + 594·15-s + 256·16-s − 414·17-s + 324·18-s + 956·19-s − 1.05e3·20-s − 1.58e3·21-s − 240·22-s + 600·23-s − 576·24-s + 1.23e3·25-s − 2.63e3·26-s − 729·27-s + 2.81e3·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.18·5-s − 0.408·6-s + 1.35·7-s + 0.353·8-s + 1/3·9-s − 0.834·10-s − 0.149·11-s − 0.288·12-s − 1.07·13-s + 0.959·14-s + 0.681·15-s + 1/4·16-s − 0.347·17-s + 0.235·18-s + 0.607·19-s − 0.590·20-s − 0.783·21-s − 0.105·22-s + 0.236·23-s − 0.204·24-s + 0.393·25-s − 0.763·26-s − 0.192·27-s + 0.678·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6\)    =    \(2 \cdot 3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{6} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.15777\)
\(L(\frac12)\)  \(\approx\)  \(1.15777\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
3 \( 1 + p^{2} T \)
good5 \( 1 + 66 T + p^{5} T^{2} \)
7 \( 1 - 176 T + p^{5} T^{2} \)
11 \( 1 + 60 T + p^{5} T^{2} \)
13 \( 1 + 658 T + p^{5} T^{2} \)
17 \( 1 + 414 T + p^{5} T^{2} \)
19 \( 1 - 956 T + p^{5} T^{2} \)
23 \( 1 - 600 T + p^{5} T^{2} \)
29 \( 1 - 5574 T + p^{5} T^{2} \)
31 \( 1 + 3592 T + p^{5} T^{2} \)
37 \( 1 + 8458 T + p^{5} T^{2} \)
41 \( 1 - 19194 T + p^{5} T^{2} \)
43 \( 1 - 13316 T + p^{5} T^{2} \)
47 \( 1 + 19680 T + p^{5} T^{2} \)
53 \( 1 + 31266 T + p^{5} T^{2} \)
59 \( 1 - 26340 T + p^{5} T^{2} \)
61 \( 1 + 31090 T + p^{5} T^{2} \)
67 \( 1 + 16804 T + p^{5} T^{2} \)
71 \( 1 - 6120 T + p^{5} T^{2} \)
73 \( 1 + 25558 T + p^{5} T^{2} \)
79 \( 1 - 74408 T + p^{5} T^{2} \)
83 \( 1 + 6468 T + p^{5} T^{2} \)
89 \( 1 + 32742 T + p^{5} T^{2} \)
97 \( 1 - 166082 T + p^{5} T^{2} \)
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

−22.52800198725495672255326514333, −21.08305845822612052852640611502, −19.56932226212139151153669468946, −17.64337214084370917772244375206, −15.87037230282555035666812175867, −14.50668450501514361821022249050, −12.21771055111024451355280485923, −11.12049523748730052768597521933, −7.61949107414732268202119369706, −4.75217914279908534590718045926, 4.75217914279908534590718045926, 7.61949107414732268202119369706, 11.12049523748730052768597521933, 12.21771055111024451355280485923, 14.50668450501514361821022249050, 15.87037230282555035666812175867, 17.64337214084370917772244375206, 19.56932226212139151153669468946, 21.08305845822612052852640611502, 22.52800198725495672255326514333

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