Properties

Degree 2
Conductor 36
Sign $1$
Self-dual yes
Motivic weight 1

Origins

Downloads

Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s + 2·13-s + 8·19-s − 5·25-s − 4·31-s − 10·37-s + 8·43-s + 9·49-s + 14·61-s − 16·67-s − 10·73-s − 4·79-s − 8·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.511·7-s + 0.554·13-s + 1.835·19-s − 25-s − 0.718·31-s − 1.643·37-s + 1.219·43-s + 1.285·49-s + 1.792·61-s − 1.954·67-s − 1.170·73-s − 0.450·79-s − 0.838·91-s + 1.421·97-s + 0.099·101-s + 0.098·103-s + 0.096·107-s + 0.095·109-s + 0.094·113-s + ⋯

Functional equation

\[\begin{align} \Lambda(s)=\mathstrut & 36 ^{s/2} \Gamma_{\C}(s) \cdot L(s)\cr =\mathstrut & \Lambda(2-s) \end{align} \]
\[\begin{align} \Lambda(s)=\mathstrut & 36 ^{s/2} \Gamma_{\C}(s+0.5) \cdot L(s)\cr =\mathstrut & \Lambda(1-s) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $1$
weight  =  1
character  :  $\chi_{36} (1, \cdot )$
Sato-Tate  :  $N(\mathrm{U}(1))$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 36,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7010910527$
$L(\frac12)$  $\approx$  $0.7010910527$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2$1$
3$1$
good5$1+5T^{2}$
7$1+4T+7T^{2}$
11$1+11T^{2}$
13$1-2T+13T^{2}$
17$1+17T^{2}$
19$1-8T+19T^{2}$
23$1+23T^{2}$
29$1+29T^{2}$
31$1+4T+31T^{2}$
37$1+10T+37T^{2}$
41$1+41T^{2}$
43$1-8T+43T^{2}$
47$1+47T^{2}$
53$1+53T^{2}$
59$1+59T^{2}$
61$1-14T+61T^{2}$
67$1+16T+67T^{2}$
71$1+71T^{2}$
73$1+10T+73T^{2}$
79$1+4T+79T^{2}$
83$1+83T^{2}$
89$1+89T^{2}$
97$1-14T+97T^{2}$
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\[\begin{equation} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{equation}\]

Imaginary part of the first few zeros on the critical line

−19.74858543777127, −18.85850998760574, −17.68732582410034, −16.25038600345249, −15.69696813163501, −13.99634105119279, −13.01055982622440, −11.77437667375268, −10.17441103098667, −9.113424945499137, −7.266467310821319, −5.802689552546196, −3.443343367909477, 3.443343367909477, 5.802689552546196, 7.266467310821319, 9.113424945499137, 10.17441103098667, 11.77437667375268, 13.01055982622440, 13.99634105119279, 15.69696813163501, 16.25038600345249, 17.68732582410034, 18.85850998760574, 19.74858543777127

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