Properties

Label 2-21780-1.1-c1-0-1
Degree $2$
Conductor $21780$
Sign $1$
Analytic cond. $173.914$
Root an. cond. $13.1876$
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  − 5-s − 4·17-s + 4·19-s − 6·23-s + 25-s + 2·29-s − 6·37-s − 10·41-s − 4·43-s − 10·47-s − 7·49-s − 2·53-s + 4·59-s + 14·61-s + 2·67-s − 4·71-s + 4·73-s + 8·79-s + 12·83-s + 4·85-s − 6·89-s − 4·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.970·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s − 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.45·47-s − 49-s − 0.274·53-s + 0.520·59-s + 1.79·61-s + 0.244·67-s − 0.474·71-s + 0.468·73-s + 0.900·79-s + 1.31·83-s + 0.433·85-s − 0.635·89-s − 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21780\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(173.914\)
Root analytic conductor: \(13.1876\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 21780,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.210347897\)
\(L(\frac12)\) \(\approx\) \(1.210347897\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good7 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.74876910872904, −15.00325578603350, −14.48909354455482, −13.93918630620643, −13.33162008326960, −12.96452572274395, −12.03701087032015, −11.84010246989381, −11.26898813833976, −10.59925921296287, −9.978859687843149, −9.562355664999373, −8.741271892772494, −8.235364278494929, −7.827139537600140, −6.808360016558293, −6.732257755695266, −5.794198301743924, −5.035637840205075, −4.620972827652827, −3.641945660505374, −3.334017110607215, −2.261793318634961, −1.612148419931644, −0.4394188709132055, 0.4394188709132055, 1.612148419931644, 2.261793318634961, 3.334017110607215, 3.641945660505374, 4.620972827652827, 5.035637840205075, 5.794198301743924, 6.732257755695266, 6.808360016558293, 7.827139537600140, 8.235364278494929, 8.741271892772494, 9.562355664999373, 9.978859687843149, 10.59925921296287, 11.26898813833976, 11.84010246989381, 12.03701087032015, 12.96452572274395, 13.33162008326960, 13.93918630620643, 14.48909354455482, 15.00325578603350, 15.74876910872904

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