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Properties

Label 2-2891-413.58-c0-0-1
Degree $2$
Conductor $2891$
Sign $0.968 - 0.250i$
Analytic cond. $1.44279$
Root an. cond. $1.20116$
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  + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)12-s + 0.999·15-s + (−0.499 + 0.866i)16-s + (1 + 1.73i)17-s + (−0.5 + 0.866i)19-s + 0.999·20-s − 27-s − 29-s + 41-s + 0.999·48-s + (0.999 − 1.73i)51-s + (0.5 + 0.866i)53-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)12-s + 0.999·15-s + (−0.499 + 0.866i)16-s + (1 + 1.73i)17-s + (−0.5 + 0.866i)19-s + 0.999·20-s − 27-s − 29-s + 41-s + 0.999·48-s + (0.999 − 1.73i)51-s + (0.5 + 0.866i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2891\)    =    \(7^{2} \cdot 59\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(1.44279\)
Root analytic conductor: \(1.20116\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2891} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2891,\ (\ :0),\ 0.968 - 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6952549881\)
\(L(\frac12)\) \(\approx\) \(0.6952549881\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
59 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 2T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.970734531638442034114979246400, −8.024103983984996056450849366681, −7.45238127489098577839609074551, −6.56336117541269695721486973867, −6.00704660709393530079812375090, −5.47614292994195526031916082798, −4.11667080419362854687182880336, −3.57013077139463056624510474976, −2.04438277647124221582061458663, −1.14196562623647906059293785993, 0.56076330969370378901960868609, 2.50869023593002593909654373793, 3.62048329926210282752191082381, 4.26795751869065964106450835222, 5.04318682543637368771412246906, 5.32302814214158257546036957263, 6.78816166163217476550127867116, 7.62739210940015870500670592921, 8.137850191310251402014417204512, 9.135966634529540245822547923456

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