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Properties

Label 2-462e2-1.1-c1-0-45
Degree $2$
Conductor $213444$
Sign $-1$
Analytic cond. $1704.35$
Root an. cond. $41.2838$
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  − 5-s − 4·13-s + 6·17-s − 2·19-s − 23-s − 4·25-s + 2·29-s + 31-s − 9·37-s − 6·41-s − 8·43-s − 8·47-s − 10·53-s + 59-s − 2·61-s + 4·65-s + 11·67-s − 11·71-s − 14·73-s + 14·79-s − 4·83-s − 6·85-s + 13·89-s + 2·95-s + 9·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.10·13-s + 1.45·17-s − 0.458·19-s − 0.208·23-s − 4/5·25-s + 0.371·29-s + 0.179·31-s − 1.47·37-s − 0.937·41-s − 1.21·43-s − 1.16·47-s − 1.37·53-s + 0.130·59-s − 0.256·61-s + 0.496·65-s + 1.34·67-s − 1.30·71-s − 1.63·73-s + 1.57·79-s − 0.439·83-s − 0.650·85-s + 1.37·89-s + 0.205·95-s + 0.913·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(213444\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1704.35\)
Root analytic conductor: \(41.2838\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 213444,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good5 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + 11 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−13.16005496177091, −12.74343477844221, −12.06605146243123, −11.97086791917910, −11.59104836033899, −10.84257412731539, −10.35007178513028, −9.913558972358996, −9.690507171214332, −8.976952066097355, −8.279946194752172, −8.174282344238226, −7.415217209101106, −7.241486864702315, −6.464194793429584, −6.116353894649280, −5.331605307117188, −4.990473098887484, −4.534444583520370, −3.777584072137577, −3.283896836940751, −2.945747419233556, −1.889849097097457, −1.730184173249911, −0.6439690283064523, 0, 0.6439690283064523, 1.730184173249911, 1.889849097097457, 2.945747419233556, 3.283896836940751, 3.777584072137577, 4.534444583520370, 4.990473098887484, 5.331605307117188, 6.116353894649280, 6.464194793429584, 7.241486864702315, 7.415217209101106, 8.174282344238226, 8.279946194752172, 8.976952066097355, 9.690507171214332, 9.913558972358996, 10.35007178513028, 10.84257412731539, 11.59104836033899, 11.97086791917910, 12.06605146243123, 12.74343477844221, 13.16005496177091

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