Properties

Label 2-12279-1.1-c1-0-5
Degree $2$
Conductor $12279$
Sign $-1$
Analytic cond. $98.0483$
Root an. cond. $9.90193$
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  − 2·2-s − 3-s + 2·4-s − 4·5-s + 2·6-s − 4·7-s + 9-s + 8·10-s − 5·11-s − 2·12-s − 6·13-s + 8·14-s + 4·15-s − 4·16-s − 7·17-s − 2·18-s − 4·19-s − 8·20-s + 4·21-s + 10·22-s − 6·23-s + 11·25-s + 12·26-s − 27-s − 8·28-s − 29-s − 8·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 1.78·5-s + 0.816·6-s − 1.51·7-s + 1/3·9-s + 2.52·10-s − 1.50·11-s − 0.577·12-s − 1.66·13-s + 2.13·14-s + 1.03·15-s − 16-s − 1.69·17-s − 0.471·18-s − 0.917·19-s − 1.78·20-s + 0.872·21-s + 2.13·22-s − 1.25·23-s + 11/5·25-s + 2.35·26-s − 0.192·27-s − 1.51·28-s − 0.185·29-s − 1.46·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12279 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12279\)    =    \(3 \cdot 4093\)
Sign: $-1$
Analytic conductor: \(98.0483\)
Root analytic conductor: \(9.90193\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((2,\ 12279,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
4093 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
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

−17.08504909404339, −16.62527713217313, −16.07343685278927, −15.81557577461617, −15.29549728521287, −14.80667033564637, −13.52087019233196, −13.00661852828197, −12.48508311235876, −11.98597935612577, −11.29891565093244, −10.62867238987283, −10.38421871099467, −9.724916744588480, −8.982129661802103, −8.487596855237308, −7.750401889248535, −7.196512176232023, −7.011164349394375, −6.110204111065486, −5.030759469618496, −4.437991017085432, −3.716083149460321, −2.724338755782939, −2.025543159355622, 0, 0, 0, 2.025543159355622, 2.724338755782939, 3.716083149460321, 4.437991017085432, 5.030759469618496, 6.110204111065486, 7.011164349394375, 7.196512176232023, 7.750401889248535, 8.487596855237308, 8.982129661802103, 9.724916744588480, 10.38421871099467, 10.62867238987283, 11.29891565093244, 11.98597935612577, 12.48508311235876, 13.00661852828197, 13.52087019233196, 14.80667033564637, 15.29549728521287, 15.81557577461617, 16.07343685278927, 16.62527713217313, 17.08504909404339

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