Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 59 $
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 + 10·5-s − 36·6-s + 144·7-s + 64·8-s + 81·9-s + 40·10-s + 668·11-s − 144·12-s − 270·13-s + 576·14-s − 90·15-s + 256·16-s − 758·17-s + 324·18-s + 868·19-s + 160·20-s − 1.29e3·21-s + 2.67e3·22-s + 784·23-s − 576·24-s − 3.02e3·25-s − 1.08e3·26-s − 729·27-s + 2.30e3·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.178·5-s − 0.408·6-s + 1.11·7-s + 0.353·8-s + 1/3·9-s + 0.126·10-s + 1.66·11-s − 0.288·12-s − 0.443·13-s + 0.785·14-s − 0.103·15-s + 1/4·16-s − 0.636·17-s + 0.235·18-s + 0.551·19-s + 0.0894·20-s − 0.641·21-s + 1.17·22-s + 0.309·23-s − 0.204·24-s − 0.967·25-s − 0.313·26-s − 0.192·27-s + 0.555·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(354\)    =    \(2 \cdot 3 \cdot 59\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{354} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 354,\ (\ :5/2),\ 1)$
$L(3)$  $\approx$  $3.678629785$
$L(\frac12)$  $\approx$  $3.678629785$
$L(\frac{7}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;59\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
3 \( 1 + p^{2} T \)
59 \( 1 + p^{2} T \)
good5 \( 1 - 2 p T + p^{5} T^{2} \)
7 \( 1 - 144 T + p^{5} T^{2} \)
11 \( 1 - 668 T + p^{5} T^{2} \)
13 \( 1 + 270 T + p^{5} T^{2} \)
17 \( 1 + 758 T + p^{5} T^{2} \)
19 \( 1 - 868 T + p^{5} T^{2} \)
23 \( 1 - 784 T + p^{5} T^{2} \)
29 \( 1 + 4574 T + p^{5} T^{2} \)
31 \( 1 - 8948 T + p^{5} T^{2} \)
37 \( 1 + 670 T + p^{5} T^{2} \)
41 \( 1 + 7934 T + p^{5} T^{2} \)
43 \( 1 - 4884 T + p^{5} T^{2} \)
47 \( 1 - 24280 T + p^{5} T^{2} \)
53 \( 1 - 28962 T + p^{5} T^{2} \)
61 \( 1 - 30490 T + p^{5} T^{2} \)
67 \( 1 + 30764 T + p^{5} T^{2} \)
71 \( 1 - 22452 T + p^{5} T^{2} \)
73 \( 1 + 20966 T + p^{5} T^{2} \)
79 \( 1 - 70520 T + p^{5} T^{2} \)
83 \( 1 - 29756 T + p^{5} T^{2} \)
89 \( 1 + 16470 T + p^{5} T^{2} \)
97 \( 1 - 18506 T + p^{5} T^{2} \)
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

−10.98160926862366538202963019933, −9.856284908109604318112157129791, −8.803735991798385176698761250993, −7.55032089973788689970718212774, −6.64447280099150232359855994146, −5.67714075603538152033125374148, −4.68608015793604134522963916646, −3.85932491596934879300895229794, −2.12776359339558979115944370653, −1.04523531135658375406310536150, 1.04523531135658375406310536150, 2.12776359339558979115944370653, 3.85932491596934879300895229794, 4.68608015793604134522963916646, 5.67714075603538152033125374148, 6.64447280099150232359855994146, 7.55032089973788689970718212774, 8.803735991798385176698761250993, 9.856284908109604318112157129791, 10.98160926862366538202963019933

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