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Properties

Label 2-28322-1.1-c1-0-14
Degree $2$
Conductor $28322$
Sign $-1$
Analytic cond. $226.152$
Root an. cond. $15.0383$
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  + 2-s − 2·3-s + 4-s − 4·5-s − 2·6-s + 8-s + 9-s − 4·10-s + 6·11-s − 2·12-s + 2·13-s + 8·15-s + 16-s + 18-s − 4·20-s + 6·22-s + 4·23-s − 2·24-s + 11·25-s + 2·26-s + 4·27-s − 8·29-s + 8·30-s + 32-s − 12·33-s + 36-s − 4·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 1.26·10-s + 1.80·11-s − 0.577·12-s + 0.554·13-s + 2.06·15-s + 1/4·16-s + 0.235·18-s − 0.894·20-s + 1.27·22-s + 0.834·23-s − 0.408·24-s + 11/5·25-s + 0.392·26-s + 0.769·27-s − 1.48·29-s + 1.46·30-s + 0.176·32-s − 2.08·33-s + 1/6·36-s − 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28322\)    =    \(2 \cdot 7^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(226.152\)
Root analytic conductor: \(15.0383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 28322,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.35518651053768, −15.10455592169132, −14.50018357990812, −13.97437740357143, −13.20575670748954, −12.56725064288563, −12.08910301271043, −11.78536701054544, −11.32301124591797, −11.00768049536377, −10.48832877272133, −9.469660981044888, −8.801195985880389, −8.395049496514515, −7.410676248661756, −7.145701028228627, −6.479494004677066, −6.034076931203600, −5.246451094384001, −4.639638250808949, −4.103260255855729, −3.559041449795808, −3.097007398285393, −1.661949598870015, −0.9325766224243432, 0, 0.9325766224243432, 1.661949598870015, 3.097007398285393, 3.559041449795808, 4.103260255855729, 4.639638250808949, 5.246451094384001, 6.034076931203600, 6.479494004677066, 7.145701028228627, 7.410676248661756, 8.395049496514515, 8.801195985880389, 9.469660981044888, 10.48832877272133, 11.00768049536377, 11.32301124591797, 11.78536701054544, 12.08910301271043, 12.56725064288563, 13.20575670748954, 13.97437740357143, 14.50018357990812, 15.10455592169132, 15.35518651053768

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