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Properties

Label 2-21168-1.1-c1-0-44
Degree $2$
Conductor $21168$
Sign $1$
Analytic cond. $169.027$
Root an. cond. $13.0010$
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  + 7·13-s + 8·19-s − 5·25-s + 11·31-s − 37-s + 13·43-s + 61-s − 11·67-s + 10·73-s + 13·79-s + 19·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.94·13-s + 1.83·19-s − 25-s + 1.97·31-s − 0.164·37-s + 1.98·43-s + 0.128·61-s − 1.34·67-s + 1.17·73-s + 1.46·79-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21168\)    =    \(2^{4} \cdot 3^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(169.027\)
Root analytic conductor: \(13.0010\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 21168,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.041417228\)
\(L(\frac12)\) \(\approx\) \(3.041417228\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + p T^{2} \) 1.5.a
11 \( 1 + p T^{2} \) 1.11.a
13 \( 1 - 7 T + p T^{2} \) 1.13.ah
17 \( 1 + p T^{2} \) 1.17.a
19 \( 1 - 8 T + p T^{2} \) 1.19.ai
23 \( 1 + p T^{2} \) 1.23.a
29 \( 1 + p T^{2} \) 1.29.a
31 \( 1 - 11 T + p T^{2} \) 1.31.al
37 \( 1 + T + p T^{2} \) 1.37.b
41 \( 1 + p T^{2} \) 1.41.a
43 \( 1 - 13 T + p T^{2} \) 1.43.an
47 \( 1 + p T^{2} \) 1.47.a
53 \( 1 + p T^{2} \) 1.53.a
59 \( 1 + p T^{2} \) 1.59.a
61 \( 1 - T + p T^{2} \) 1.61.ab
67 \( 1 + 11 T + p T^{2} \) 1.67.l
71 \( 1 + p T^{2} \) 1.71.a
73 \( 1 - 10 T + p T^{2} \) 1.73.ak
79 \( 1 - 13 T + p T^{2} \) 1.79.an
83 \( 1 + p T^{2} \) 1.83.a
89 \( 1 + p T^{2} \) 1.89.a
97 \( 1 - 19 T + p T^{2} \) 1.97.at
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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

−15.69466643980113, −15.27170364430101, −14.29960615510944, −13.87541060919501, −13.53035167367578, −13.04456432629155, −12.07490493598606, −11.86615458193025, −11.19022839702959, −10.67727583602905, −10.08589465851105, −9.351282134932699, −9.030382425377664, −8.094540820215498, −7.911626493914281, −7.082395822335311, −6.318360922404466, −5.913537243787103, −5.281461070328148, −4.442376363190028, −3.766939207339082, −3.220306593318979, −2.407403007941167, −1.329861266233467, −0.8173126364220043, 0.8173126364220043, 1.329861266233467, 2.407403007941167, 3.220306593318979, 3.766939207339082, 4.442376363190028, 5.281461070328148, 5.913537243787103, 6.318360922404466, 7.082395822335311, 7.911626493914281, 8.094540820215498, 9.030382425377664, 9.351282134932699, 10.08589465851105, 10.67727583602905, 11.19022839702959, 11.86615458193025, 12.07490493598606, 13.04456432629155, 13.53035167367578, 13.87541060919501, 14.29960615510944, 15.27170364430101, 15.69466643980113

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