Properties

Label 2-25872-1.1-c1-0-3
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 + 2·13-s − 2·15-s − 4·17-s − 6·19-s − 25-s + 27-s − 8·29-s − 8·31-s − 33-s + 10·37-s + 2·39-s − 8·41-s + 2·43-s − 2·45-s − 8·47-s − 4·51-s − 2·53-s + 2·55-s − 6·57-s + 12·59-s − 10·61-s − 4·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.516·15-s − 0.970·17-s − 1.37·19-s − 1/5·25-s + 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.174·33-s + 1.64·37-s + 0.320·39-s − 1.24·41-s + 0.304·43-s − 0.298·45-s − 1.16·47-s − 0.560·51-s − 0.274·53-s + 0.269·55-s − 0.794·57-s + 1.56·59-s − 1.28·61-s − 0.496·65-s − 1.46·67-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

\(L(1)\) \(\approx\) \(1.003998050\)
\(L(\frac12)\) \(\approx\) \(1.003998050\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−15.16326380357521, −14.87813915775571, −14.52803511436998, −13.50964903361713, −13.09268940134022, −12.97369739106917, −12.02644188493257, −11.53456166660604, −10.93205166626398, −10.63706444113735, −9.793130170889716, −9.113184373411291, −8.745550606297384, −8.122087642021888, −7.617859731737542, −7.129917196125340, −6.329280283777802, −5.848904660662853, −4.844101429850706, −4.316210959903267, −3.727197067048951, −3.211475296055988, −2.195440365873814, −1.733427515058483, −0.3624833271469523, 0.3624833271469523, 1.733427515058483, 2.195440365873814, 3.211475296055988, 3.727197067048951, 4.316210959903267, 4.844101429850706, 5.848904660662853, 6.329280283777802, 7.129917196125340, 7.617859731737542, 8.122087642021888, 8.745550606297384, 9.113184373411291, 9.793130170889716, 10.63706444113735, 10.93205166626398, 11.53456166660604, 12.02644188493257, 12.97369739106917, 13.09268940134022, 13.50964903361713, 14.52803511436998, 14.87813915775571, 15.16326380357521

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