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Properties

Label 2-25872-1.1-c1-0-6
Degree $2$
Conductor $25872$
Sign $1$
Analytic cond. $206.588$
Root an. cond. $14.3732$
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  − 3-s − 2·5-s + 9-s − 11-s + 6·13-s + 2·15-s + 2·17-s − 8·19-s + 4·23-s − 25-s − 27-s + 2·29-s − 8·31-s + 33-s + 6·37-s − 6·39-s + 2·41-s − 8·43-s − 2·45-s − 4·47-s − 2·51-s + 2·53-s + 2·55-s + 8·57-s − 12·59-s − 10·61-s − 12·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s + 1.66·13-s + 0.516·15-s + 0.485·17-s − 1.83·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.174·33-s + 0.986·37-s − 0.960·39-s + 0.312·41-s − 1.21·43-s − 0.298·45-s − 0.583·47-s − 0.280·51-s + 0.274·53-s + 0.269·55-s + 1.05·57-s − 1.56·59-s − 1.28·61-s − 1.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25872\)    =    \(2^{4} \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(206.588\)
Root analytic conductor: \(14.3732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 25872,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.043076376\)
\(L(\frac12)\) \(\approx\) \(1.043076376\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.29194121820358, −15.04750102609638, −14.32243900086530, −13.60661869621723, −12.99455889788006, −12.73082749407298, −12.08198376199042, −11.39017095499119, −11.00194576512832, −10.72832443089137, −10.01057309923789, −9.189951311445592, −8.666300375385259, −8.091281586485820, −7.644091537991143, −6.859083914263956, −6.255505818557304, −5.885897306751505, −4.971600178330110, −4.454066746540874, −3.687044162281581, −3.356869334483078, −2.197470018437522, −1.365786275977429, −0.4405979627844021, 0.4405979627844021, 1.365786275977429, 2.197470018437522, 3.356869334483078, 3.687044162281581, 4.454066746540874, 4.971600178330110, 5.885897306751505, 6.255505818557304, 6.859083914263956, 7.644091537991143, 8.091281586485820, 8.666300375385259, 9.189951311445592, 10.01057309923789, 10.72832443089137, 11.00194576512832, 11.39017095499119, 12.08198376199042, 12.73082749407298, 12.99455889788006, 13.60661869621723, 14.32243900086530, 15.04750102609638, 15.29194121820358

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