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Properties

Label 2-28322-1.1-c1-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s − 2·9-s + 2·10-s − 2·11-s − 12-s − 4·13-s − 2·15-s + 16-s − 2·18-s + 19-s + 2·20-s − 2·22-s − 24-s − 25-s − 4·26-s + 5·27-s − 3·29-s − 2·30-s − 7·31-s + 32-s + 2·33-s − 2·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.632·10-s − 0.603·11-s − 0.288·12-s − 1.10·13-s − 0.516·15-s + 1/4·16-s − 0.471·18-s + 0.229·19-s + 0.447·20-s − 0.426·22-s − 0.204·24-s − 1/5·25-s − 0.784·26-s + 0.962·27-s − 0.557·29-s − 0.365·30-s − 1.25·31-s + 0.176·32-s + 0.348·33-s − 1/3·36-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: \(0\)
Selberg data: \((2,\ 28322,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.913414833\)
\(L(\frac12)\) \(\approx\) \(1.913414833\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−15.09769410753148, −14.57380588050738, −14.13980376442360, −13.65015703547104, −13.02972548327196, −12.63653791144615, −12.05217972039802, −11.46234593378901, −11.09532589297973, −10.29311760035415, −10.03265416666066, −9.313052239221081, −8.719816174653258, −7.872756998283853, −7.399610772698159, −6.650569688489337, −6.159390058955539, −5.496411261338271, −5.157278852235680, −4.720276950026429, −3.615041939934084, −3.077304701514490, −2.200776374391463, −1.818584197094994, −0.4582488562238891, 0.4582488562238891, 1.818584197094994, 2.200776374391463, 3.077304701514490, 3.615041939934084, 4.720276950026429, 5.157278852235680, 5.496411261338271, 6.159390058955539, 6.650569688489337, 7.399610772698159, 7.872756998283853, 8.719816174653258, 9.313052239221081, 10.03265416666066, 10.29311760035415, 11.09532589297973, 11.46234593378901, 12.05217972039802, 12.63653791144615, 13.02972548327196, 13.65015703547104, 14.13980376442360, 14.57380588050738, 15.09769410753148

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