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Properties

Label 2-2891-413.235-c0-0-2
Degree $2$
Conductor $2891$
Sign $0.386 - 0.922i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2891\)    =    \(7^{2} \cdot 59\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(1.44279\)
Root analytic conductor: \(1.20116\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2891} (2713, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2891,\ (\ :0),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.320521024\)
\(L(\frac12)\) \(\approx\) \(1.320521024\)
\(L(1)\) 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

−8.823573607109316093543880801994, −8.239776327952519283819585513749, −7.68672783402161022958912402917, −6.88335806846954413555967232607, −6.35369637138782202490704415144, −5.30286605319036745147130926708, −4.12937392474560267474951653692, −3.42596919566228130836220305286, −2.44134576684263061045762235445, −1.72631009811630911762167908708, 0.799467565213600756285561548217, 2.09395922604873230202677430274, 3.25189859017203059997583315054, 4.33709686595721967412685829130, 4.93845402367619572127930918241, 5.35434777230991222905767567092, 6.46750471128710390688533075515, 7.25128782032203409565381611584, 8.502879611332539410857679956041, 9.151552720672709293823699975324

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