Properties

Degree $2$
Conductor $28322$
Sign $-1$
Motivic weight $1$
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 − 2·6-s + 8-s + 9-s − 6·11-s − 2·12-s − 2·13-s + 16-s + 18-s + 4·19-s − 6·22-s − 2·24-s − 5·25-s − 2·26-s + 4·27-s − 4·31-s + 32-s + 12·33-s + 36-s + 4·37-s + 4·38-s + 4·39-s + 6·41-s + 8·43-s − 6·44-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.577·12-s − 0.554·13-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 1.27·22-s − 0.408·24-s − 25-s − 0.392·26-s + 0.769·27-s − 0.718·31-s + 0.176·32-s + 2.08·33-s + 1/6·36-s + 0.657·37-s + 0.648·38-s + 0.640·39-s + 0.937·41-s + 1.21·43-s − 0.904·44-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{28322} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−15.62409703535378, −15.04205630843256, −14.19917467229632, −14.00883159843997, −13.11601907067090, −12.80742795298601, −12.35698114775160, −11.69907118063491, −11.24841503617883, −10.83382041501906, −10.23877217507859, −9.767855378236961, −9.028329174338571, −8.060950817464079, −7.581014872933697, −7.235998692806199, −6.258703329131722, −5.836049490258861, −5.351566171254877, −4.904474632161383, −4.299976143335690, −3.363295403306189, −2.684192714115655, −2.070233304346154, −0.8623572496243492, 0, 0.8623572496243492, 2.070233304346154, 2.684192714115655, 3.363295403306189, 4.299976143335690, 4.904474632161383, 5.351566171254877, 5.836049490258861, 6.258703329131722, 7.235998692806199, 7.581014872933697, 8.060950817464079, 9.028329174338571, 9.767855378236961, 10.23877217507859, 10.83382041501906, 11.24841503617883, 11.69907118063491, 12.35698114775160, 12.80742795298601, 13.11601907067090, 14.00883159843997, 14.19917467229632, 15.04205630843256, 15.62409703535378

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