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Properties

Label 2-28322-1.1-c1-0-34
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 + 3·3-s + 4-s + 2·5-s − 3·6-s − 8-s + 6·9-s − 2·10-s − 2·11-s + 3·12-s + 6·13-s + 6·15-s + 16-s − 6·18-s − 7·19-s + 2·20-s + 2·22-s − 8·23-s − 3·24-s − 25-s − 6·26-s + 9·27-s − 9·29-s − 6·30-s − 5·31-s − 32-s − 6·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.894·5-s − 1.22·6-s − 0.353·8-s + 2·9-s − 0.632·10-s − 0.603·11-s + 0.866·12-s + 1.66·13-s + 1.54·15-s + 1/4·16-s − 1.41·18-s − 1.60·19-s + 0.447·20-s + 0.426·22-s − 1.66·23-s − 0.612·24-s − 1/5·25-s − 1.17·26-s + 1.73·27-s − 1.67·29-s − 1.09·30-s − 0.898·31-s − 0.176·32-s − 1.04·33-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 \( 1 - p T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.49797192042283, −14.77244647201287, −14.61508558349075, −13.73992703316511, −13.46419579480240, −13.06755954861796, −12.53214155722838, −11.61596474016294, −10.85600241368957, −10.48615322651662, −9.871547899885416, −9.455027936945859, −8.713128529653583, −8.629491503278164, −7.920195493051197, −7.511986355787370, −6.711704295768907, −5.969939526031034, −5.702503626528387, −4.381492977807523, −3.833057026705117, −3.276310086024334, −2.373156239640037, −1.902348316889352, −1.522232852551430, 0, 1.522232852551430, 1.902348316889352, 2.373156239640037, 3.276310086024334, 3.833057026705117, 4.381492977807523, 5.702503626528387, 5.969939526031034, 6.711704295768907, 7.511986355787370, 7.920195493051197, 8.629491503278164, 8.713128529653583, 9.455027936945859, 9.871547899885416, 10.48615322651662, 10.85600241368957, 11.61596474016294, 12.53214155722838, 13.06755954861796, 13.46419579480240, 13.73992703316511, 14.61508558349075, 14.77244647201287, 15.49797192042283

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