Properties

Label 2-21780-1.1-c1-0-15
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s − 2·13-s − 2·19-s + 25-s + 8·31-s − 2·35-s + 2·37-s − 2·43-s − 3·49-s − 6·53-s + 12·59-s − 2·61-s − 2·65-s − 4·67-s − 2·73-s + 10·79-s − 12·83-s + 6·89-s + 4·91-s − 2·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.554·13-s − 0.458·19-s + 1/5·25-s + 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.304·43-s − 3/7·49-s − 0.824·53-s + 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.488·67-s − 0.234·73-s + 1.12·79-s − 1.31·83-s + 0.635·89-s + 0.419·91-s − 0.205·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

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

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: $\chi_{21780} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 21780,\ (\ :1/2),\ -1)\)

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

−15.84527238612872, −15.31058807020732, −14.66515184361647, −14.27303558210137, −13.51206307558259, −13.18520238580969, −12.60438941263597, −12.07245193373579, −11.49711994675907, −10.81723193535146, −10.11580682098118, −9.870265197619914, −9.235313990841747, −8.618742346115355, −7.992474624899778, −7.330122705893078, −6.546245288903881, −6.325330513022705, −5.511751432466468, −4.844701860852199, −4.216867700592509, −3.361792214262151, −2.725191039930622, −2.059299741342764, −1.043403669405913, 0, 1.043403669405913, 2.059299741342764, 2.725191039930622, 3.361792214262151, 4.216867700592509, 4.844701860852199, 5.511751432466468, 6.325330513022705, 6.546245288903881, 7.330122705893078, 7.992474624899778, 8.618742346115355, 9.235313990841747, 9.870265197619914, 10.11580682098118, 10.81723193535146, 11.49711994675907, 12.07245193373579, 12.60438941263597, 13.18520238580969, 13.51206307558259, 14.27303558210137, 14.66515184361647, 15.31058807020732, 15.84527238612872

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