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Properties

Label 2-2592-216.115-c0-0-0
Degree $2$
Conductor $2592$
Sign $0.286 + 0.957i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.266 − 1.50i)11-s + (−0.939 − 1.62i)17-s + (0.173 − 0.300i)19-s + (−0.939 + 0.342i)25-s + (0.326 + 0.118i)41-s + (0.326 − 1.85i)43-s + (0.173 + 0.984i)49-s + (0.0603 + 0.342i)59-s + (1.43 + 0.524i)67-s + (0.939 − 1.62i)73-s + (0.939 − 0.342i)83-s + (−0.5 + 0.866i)89-s + (0.266 − 1.50i)97-s − 1.87·107-s + (0.173 + 0.984i)113-s + ⋯
L(s)  = 1  + (0.266 − 1.50i)11-s + (−0.939 − 1.62i)17-s + (0.173 − 0.300i)19-s + (−0.939 + 0.342i)25-s + (0.326 + 0.118i)41-s + (0.326 − 1.85i)43-s + (0.173 + 0.984i)49-s + (0.0603 + 0.342i)59-s + (1.43 + 0.524i)67-s + (0.939 − 1.62i)73-s + (0.939 − 0.342i)83-s + (−0.5 + 0.866i)89-s + (0.266 − 1.50i)97-s − 1.87·107-s + (0.173 + 0.984i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.286 + 0.957i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (2287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 0.286 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.064594478\)
\(L(\frac12)\) \(\approx\) \(1.064594478\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.173 - 0.984i)T^{2} \)
11 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
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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

−9.055031494142990505486866817635, −8.195954861598393081586011395296, −7.36923343971722127198121211326, −6.63117806527138844640263233781, −5.78892839747735628942583267218, −5.08013887555037842946441858949, −4.05614778269268557633023384366, −3.18359856974288113098183845063, −2.25770758300260994367170359302, −0.70077204607085848314467797144, 1.61734870254295898889064221325, 2.38970647387330603299013254435, 3.82295588319089525710229567989, 4.32884160958036496957294819878, 5.30184502328753100283474407153, 6.29810898411552904724742676517, 6.82272473198486642902778378127, 7.81469012273646826539891393494, 8.332071920548210663621283625020, 9.364072575651952023481903907141

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