Properties

Label 2-6e2-1.1-c1-0-0
Degree $2$
Conductor $36$
Sign $1$
Analytic cond. $0.287461$
Root an. cond. $0.536154$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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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 + 20·103-s + 2·109-s + ⋯
L(s)  = 1  − 1.51·7-s + 0.554·13-s + 1.83·19-s − 25-s − 0.718·31-s − 1.64·37-s + 1.21·43-s + 9/7·49-s + 1.79·61-s − 1.95·67-s − 1.17·73-s − 0.450·79-s − 0.838·91-s + 1.42·97-s + 1.97·103-s + 0.191·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(0.287461\)
Root analytic conductor: \(0.536154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{36} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7010910526\)
\(L(\frac12)\) \(\approx\) \(0.7010910526\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.25038600345248605266882134733, −15.69696813163500519905091038725, −13.99634105119279079569568395122, −13.01055982622439917603548235428, −11.77437667375267836950691610862, −10.17441103098667470227930585744, −9.113424945499136957715665264292, −7.26646731082131852272265775116, −5.80268955254619590131632024787, −3.44334336790947687892993758586, 3.44334336790947687892993758586, 5.80268955254619590131632024787, 7.26646731082131852272265775116, 9.113424945499136957715665264292, 10.17441103098667470227930585744, 11.77437667375267836950691610862, 13.01055982622439917603548235428, 13.99634105119279079569568395122, 15.69696813163500519905091038725, 16.25038600345248605266882134733

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