Properties

Label 2-60e2-1.1-c1-0-36
Degree $2$
Conductor $3600$
Sign $-1$
Analytic cond. $28.7461$
Root an. cond. $5.36154$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 4·11-s − 13-s − 4·17-s − 19-s − 4·23-s − 4·29-s + 5·31-s − 6·37-s − 12·41-s − 5·43-s + 8·47-s − 6·49-s + 12·53-s + 8·59-s + 7·61-s − 13·67-s + 12·71-s − 6·73-s − 4·77-s − 12·79-s − 8·83-s + 91-s − 13·97-s − 12·101-s + 4·103-s + 12·107-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.20·11-s − 0.277·13-s − 0.970·17-s − 0.229·19-s − 0.834·23-s − 0.742·29-s + 0.898·31-s − 0.986·37-s − 1.87·41-s − 0.762·43-s + 1.16·47-s − 6/7·49-s + 1.64·53-s + 1.04·59-s + 0.896·61-s − 1.58·67-s + 1.42·71-s − 0.702·73-s − 0.455·77-s − 1.35·79-s − 0.878·83-s + 0.104·91-s − 1.31·97-s − 1.19·101-s + 0.394·103-s + 1.16·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(28.7461\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−8.480609091215779589704005764301, −7.21721465207835896557438301071, −6.76273428051938626599131220880, −6.05563206904210914262529145041, −5.14876439760747994500609547156, −4.18926062477986906335244250532, −3.61547866934288568708632555501, −2.46379184444444924412745877337, −1.50367970955387668148868767413, 0, 1.50367970955387668148868767413, 2.46379184444444924412745877337, 3.61547866934288568708632555501, 4.18926062477986906335244250532, 5.14876439760747994500609547156, 6.05563206904210914262529145041, 6.76273428051938626599131220880, 7.21721465207835896557438301071, 8.480609091215779589704005764301

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