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Properties

Label 2-25992-1.1-c1-0-3
Degree $2$
Conductor $25992$
Sign $1$
Analytic cond. $207.547$
Root an. cond. $14.4064$
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  + 2·5-s − 4·11-s + 2·13-s − 2·17-s + 8·23-s − 25-s + 6·29-s − 8·31-s − 6·37-s − 6·41-s + 4·43-s − 7·49-s − 2·53-s − 8·55-s + 4·59-s − 2·61-s + 4·65-s + 4·67-s + 8·71-s + 10·73-s + 8·79-s + 4·83-s − 4·85-s − 6·89-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 49-s − 0.274·53-s − 1.07·55-s + 0.520·59-s − 0.256·61-s + 0.496·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.900·79-s + 0.439·83-s − 0.433·85-s − 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25992\)    =    \(2^{3} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(207.547\)
Root analytic conductor: \(14.4064\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 25992,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.285099802\)
\(L(\frac12)\) \(\approx\) \(2.285099802\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.43074471690596, −14.80318716278194, −14.12352758490346, −13.64892836541402, −13.24575570310661, −12.73423289439720, −12.29715978358621, −11.30535937475635, −10.97446526148068, −10.46018749640959, −9.903934544478440, −9.282555174963753, −8.783774303784158, −8.189315765477712, −7.561660573902019, −6.764693822119382, −6.466902292286386, −5.464015623981355, −5.289280900274339, −4.593909351421417, −3.603154262727234, −3.006058627814732, −2.243899609691525, −1.624942915732200, −0.5850176229063715, 0.5850176229063715, 1.624942915732200, 2.243899609691525, 3.006058627814732, 3.603154262727234, 4.593909351421417, 5.289280900274339, 5.464015623981355, 6.466902292286386, 6.764693822119382, 7.561660573902019, 8.189315765477712, 8.783774303784158, 9.282555174963753, 9.903934544478440, 10.46018749640959, 10.97446526148068, 11.30535937475635, 12.29715978358621, 12.73423289439720, 13.24575570310661, 13.64892836541402, 14.12352758490346, 14.80318716278194, 15.43074471690596

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