Properties

Label 2-23232-1.1-c1-0-72
Degree $2$
Conductor $23232$
Sign $-1$
Analytic cond. $185.508$
Root an. cond. $13.6201$
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  − 3-s + 2·5-s + 9-s − 2·13-s − 2·15-s − 2·17-s + 4·19-s + 8·23-s − 25-s − 27-s + 6·29-s − 8·31-s − 6·37-s + 2·39-s + 6·41-s − 4·43-s + 2·45-s − 7·49-s + 2·51-s + 2·53-s − 4·57-s + 4·59-s − 2·61-s − 4·65-s − 4·67-s − 8·69-s − 8·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.298·45-s − 49-s + 0.280·51-s + 0.274·53-s − 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.496·65-s − 0.488·67-s − 0.963·69-s − 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23232\)    =    \(2^{6} \cdot 3 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(185.508\)
Root analytic conductor: \(13.6201\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 23232,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
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

−15.82990475860147, −15.17216014977464, −14.64276871927380, −14.09392465953872, −13.51533669850880, −13.02760940167935, −12.57476071682863, −11.87860951167232, −11.39509703585013, −10.78638377778716, −10.27740290449854, −9.703161057728960, −9.199021555216559, −8.696453798989065, −7.820487550732530, −7.046191741606810, −6.863092616431083, −5.941952593884288, −5.490245924050042, −4.944548997435659, −4.339902203589440, −3.320450658683481, −2.711366608873740, −1.809570618965464, −1.142034592903933, 0, 1.142034592903933, 1.809570618965464, 2.711366608873740, 3.320450658683481, 4.339902203589440, 4.944548997435659, 5.490245924050042, 5.941952593884288, 6.863092616431083, 7.046191741606810, 7.820487550732530, 8.696453798989065, 9.199021555216559, 9.703161057728960, 10.27740290449854, 10.78638377778716, 11.39509703585013, 11.87860951167232, 12.57476071682863, 13.02760940167935, 13.51533669850880, 14.09392465953872, 14.64276871927380, 15.17216014977464, 15.82990475860147

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