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Properties

Label 2-231e2-1.1-c1-0-25
Degree $2$
Conductor $53361$
Sign $1$
Analytic cond. $426.089$
Root an. cond. $20.6419$
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  − 2·4-s + 7·13-s + 4·16-s + 7·19-s − 5·25-s − 7·31-s − 37-s − 5·43-s − 14·52-s − 14·61-s − 8·64-s + 11·67-s + 7·73-s − 14·76-s + 13·79-s + 14·97-s + 10·100-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 4-s + 1.94·13-s + 16-s + 1.60·19-s − 25-s − 1.25·31-s − 0.164·37-s − 0.762·43-s − 1.94·52-s − 1.79·61-s − 64-s + 1.34·67-s + 0.819·73-s − 1.60·76-s + 1.46·79-s + 1.42·97-s + 100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53361\)    =    \(3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(426.089\)
Root analytic conductor: \(20.6419\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 53361,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.929179692\)
\(L(\frac12)\) \(\approx\) \(1.929179692\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
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

−14.13441648588005, −13.97463498585902, −13.39106347732925, −13.18742655857534, −12.43343870290651, −11.93439240740147, −11.35428752434776, −10.83571373290241, −10.35901092530654, −9.510184376270986, −9.398171300098500, −8.774702314544839, −8.124570924516123, −7.857904164334617, −7.104043600099352, −6.375099663447473, −5.766017054476075, −5.397000691292517, −4.754292626500520, −3.949982792228326, −3.557216817094184, −3.133979572719027, −1.902152839083103, −1.272778341448732, −0.5422969213559723, 0.5422969213559723, 1.272778341448732, 1.902152839083103, 3.133979572719027, 3.557216817094184, 3.949982792228326, 4.754292626500520, 5.397000691292517, 5.766017054476075, 6.375099663447473, 7.104043600099352, 7.857904164334617, 8.124570924516123, 8.774702314544839, 9.398171300098500, 9.510184376270986, 10.35901092530654, 10.83571373290241, 11.35428752434776, 11.93439240740147, 12.43343870290651, 13.18742655857534, 13.39106347732925, 13.97463498585902, 14.13441648588005

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