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Properties

Label 2-27746-1.1-c1-0-0
Degree $2$
Conductor $27746$
Sign $-1$
Analytic cond. $221.552$
Root an. cond. $14.8846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $3$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s − 4·7-s − 8-s + 9-s + 2·10-s − 5·11-s − 2·12-s − 4·13-s + 4·14-s + 4·15-s + 16-s − 5·17-s − 18-s − 7·19-s − 2·20-s + 8·21-s + 5·22-s − 6·23-s + 2·24-s − 25-s + 4·26-s + 4·27-s − 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.50·11-s − 0.577·12-s − 1.10·13-s + 1.06·14-s + 1.03·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s − 1.60·19-s − 0.447·20-s + 1.74·21-s + 1.06·22-s − 1.25·23-s + 0.408·24-s − 1/5·25-s + 0.784·26-s + 0.769·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27746 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27746 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27746\)    =    \(2 \cdot 13873\)
Sign: $-1$
Analytic conductor: \(221.552\)
Root analytic conductor: \(14.8846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((2,\ 27746,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
13873 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 17 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.12105122100236, −15.62672175215584, −15.32663938139735, −14.75709483593525, −13.74103795600994, −12.94140022501254, −12.88999660930336, −12.13099810788970, −11.72551641705431, −11.21264360073354, −10.47399424479514, −10.21470812793164, −9.812160480785096, −8.881053431280070, −8.352964751307357, −7.830739244752575, −7.004131145619064, −6.749635847713361, −6.041598614522246, −5.596399542678073, −4.647193727318560, −4.259800032494830, −3.186080799199416, −2.653535485998504, −1.855901548368396, 0, 0, 0, 1.855901548368396, 2.653535485998504, 3.186080799199416, 4.259800032494830, 4.647193727318560, 5.596399542678073, 6.041598614522246, 6.749635847713361, 7.004131145619064, 7.830739244752575, 8.352964751307357, 8.881053431280070, 9.812160480785096, 10.21470812793164, 10.47399424479514, 11.21264360073354, 11.72551641705431, 12.13099810788970, 12.88999660930336, 12.94140022501254, 13.74103795600994, 14.75709483593525, 15.32663938139735, 15.62672175215584, 16.12105122100236

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