Properties

Degree $2$
Conductor $14700$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 5·11-s + 2·13-s + 2·17-s − 6·19-s + 23-s − 27-s + 3·29-s − 4·31-s − 5·33-s − 5·37-s − 2·39-s − 4·41-s − 7·43-s + 10·47-s − 2·51-s + 2·53-s + 6·57-s − 10·59-s − 8·61-s + 7·67-s − 69-s − 3·71-s − 2·73-s − 11·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.50·11-s + 0.554·13-s + 0.485·17-s − 1.37·19-s + 0.208·23-s − 0.192·27-s + 0.557·29-s − 0.718·31-s − 0.870·33-s − 0.821·37-s − 0.320·39-s − 0.624·41-s − 1.06·43-s + 1.45·47-s − 0.280·51-s + 0.274·53-s + 0.794·57-s − 1.30·59-s − 1.02·61-s + 0.855·67-s − 0.120·69-s − 0.356·71-s − 0.234·73-s − 1.23·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14700\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{14700} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 14700,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−16.60246460375253, −15.76649804417081, −15.28918190384224, −14.73243819333832, −14.08808665012491, −13.68623843428486, −12.82086065969229, −12.42524945998085, −11.83366450813506, −11.34905068861642, −10.69678729481783, −10.24246959759356, −9.506910168949557, −8.787216376676440, −8.521416278241196, −7.525415357408664, −6.858452707553713, −6.394252495219791, −5.836487641382266, −5.078521282260279, −4.241196506995553, −3.834171874144027, −2.954054394569227, −1.778982927364904, −1.221813131386276, 0, 1.221813131386276, 1.778982927364904, 2.954054394569227, 3.834171874144027, 4.241196506995553, 5.078521282260279, 5.836487641382266, 6.394252495219791, 6.858452707553713, 7.525415357408664, 8.521416278241196, 8.787216376676440, 9.506910168949557, 10.24246959759356, 10.69678729481783, 11.34905068861642, 11.83366450813506, 12.42524945998085, 12.82086065969229, 13.68623843428486, 14.08808665012491, 14.73243819333832, 15.28918190384224, 15.76649804417081, 16.60246460375253

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