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Properties

Label 2-11271-1.1-c1-0-2
Degree $2$
Conductor $11271$
Sign $1$
Analytic cond. $89.9993$
Root an. cond. $9.48680$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s + 4·5-s − 6-s + 3·8-s + 9-s − 4·10-s − 12-s − 13-s + 4·15-s − 16-s − 18-s + 4·19-s − 4·20-s + 4·23-s + 3·24-s + 11·25-s + 26-s + 27-s + 8·29-s − 4·30-s + 8·31-s − 5·32-s − 36-s + 4·37-s − 4·38-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 1.78·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 1.26·10-s − 0.288·12-s − 0.277·13-s + 1.03·15-s − 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.894·20-s + 0.834·23-s + 0.612·24-s + 11/5·25-s + 0.196·26-s + 0.192·27-s + 1.48·29-s − 0.730·30-s + 1.43·31-s − 0.883·32-s − 1/6·36-s + 0.657·37-s − 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11271 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11271 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11271\)    =    \(3 \cdot 13 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(89.9993\)
Root analytic conductor: \(9.48680\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 11271,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.754048791\)
\(L(\frac12)\) \(\approx\) \(2.754048791\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−16.81058834419264, −16.00240763730250, −15.32031713033411, −14.56228438031993, −13.91487745532691, −13.82652181795706, −13.24362253229674, −12.64971210122048, −11.94213751469863, −10.92992623256439, −10.30590643383315, −9.909930124004838, −9.501190639394691, −8.888752900826383, −8.472242696739410, −7.666993739351871, −6.975696656562531, −6.300251987633887, −5.499279558410418, −4.897573518923026, −4.263549178564705, −2.984182917893370, −2.540653955305343, −1.469347423157143, −0.9467243748400541, 0.9467243748400541, 1.469347423157143, 2.540653955305343, 2.984182917893370, 4.263549178564705, 4.897573518923026, 5.499279558410418, 6.300251987633887, 6.975696656562531, 7.666993739351871, 8.472242696739410, 8.888752900826383, 9.501190639394691, 9.909930124004838, 10.30590643383315, 10.92992623256439, 11.94213751469863, 12.64971210122048, 13.24362253229674, 13.82652181795706, 13.91487745532691, 14.56228438031993, 15.32031713033411, 16.00240763730250, 16.81058834419264

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