Properties

Label 2-163e2-1.1-c1-0-1
Degree $2$
Conductor $26569$
Sign $-1$
Analytic cond. $212.154$
Root an. cond. $14.5655$
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  − 2·4-s − 3·9-s + 4·16-s − 5·25-s + 6·36-s − 41-s + 3·43-s + 5·47-s − 7·49-s + 7·53-s − 9·61-s − 8·64-s + 11·71-s + 9·81-s + 13·83-s − 15·97-s + 10·100-s + ⋯
L(s)  = 1  − 4-s − 9-s + 16-s − 25-s + 36-s − 0.156·41-s + 0.457·43-s + 0.729·47-s − 49-s + 0.961·53-s − 1.15·61-s − 64-s + 1.30·71-s + 81-s + 1.42·83-s − 1.52·97-s + 100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26569\)    =    \(163^{2}\)
Sign: $-1$
Analytic conductor: \(212.154\)
Root analytic conductor: \(14.5655\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 26569,\ (\ :1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad163 \( 1 \)
good2 \( 1 + p T^{2} \) 1.2.a
3 \( 1 + p T^{2} \) 1.3.a
5 \( 1 + p T^{2} \) 1.5.a
7 \( 1 + p T^{2} \) 1.7.a
11 \( 1 + p T^{2} \) 1.11.a
13 \( 1 + p T^{2} \) 1.13.a
17 \( 1 + p T^{2} \) 1.17.a
19 \( 1 + p T^{2} \) 1.19.a
23 \( 1 + p T^{2} \) 1.23.a
29 \( 1 + p T^{2} \) 1.29.a
31 \( 1 + p T^{2} \) 1.31.a
37 \( 1 + p T^{2} \) 1.37.a
41 \( 1 + T + p T^{2} \) 1.41.b
43 \( 1 - 3 T + p T^{2} \) 1.43.ad
47 \( 1 - 5 T + p T^{2} \) 1.47.af
53 \( 1 - 7 T + p T^{2} \) 1.53.ah
59 \( 1 + p T^{2} \) 1.59.a
61 \( 1 + 9 T + p T^{2} \) 1.61.j
67 \( 1 + p T^{2} \) 1.67.a
71 \( 1 - 11 T + p T^{2} \) 1.71.al
73 \( 1 + p T^{2} \) 1.73.a
79 \( 1 + p T^{2} \) 1.79.a
83 \( 1 - 13 T + p T^{2} \) 1.83.an
89 \( 1 + p T^{2} \) 1.89.a
97 \( 1 + 15 T + p T^{2} \) 1.97.p
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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

−15.32289965722446, −15.06437654539026, −14.33306698836764, −13.93794556700604, −13.56264499895653, −12.97424067964289, −12.27631954712754, −11.95545378602271, −11.18359415701897, −10.73891318546896, −9.999162301866341, −9.511895801018704, −8.990481511019226, −8.453382841430840, −7.946403877187488, −7.405364290693415, −6.474285949177591, −5.893743973482476, −5.377409030875712, −4.785077507040121, −4.043287233078329, −3.493840680195974, −2.757533966348102, −1.905032829689204, −0.8347343620222809, 0, 0.8347343620222809, 1.905032829689204, 2.757533966348102, 3.493840680195974, 4.043287233078329, 4.785077507040121, 5.377409030875712, 5.893743973482476, 6.474285949177591, 7.405364290693415, 7.946403877187488, 8.453382841430840, 8.990481511019226, 9.511895801018704, 9.999162301866341, 10.73891318546896, 11.18359415701897, 11.95545378602271, 12.27631954712754, 12.97424067964289, 13.56264499895653, 13.93794556700604, 14.33306698836764, 15.06437654539026, 15.32289965722446

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