Properties

Degree $2$
Conductor $16650$
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 + 4-s + 8-s − 4·11-s + 2·13-s + 16-s + 2·17-s + 4·19-s − 4·22-s − 8·23-s + 2·26-s + 2·29-s + 8·31-s + 32-s + 2·34-s − 37-s + 4·38-s − 10·41-s − 12·43-s − 4·44-s − 8·46-s − 7·49-s + 2·52-s + 6·53-s + 2·58-s − 4·59-s − 10·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.852·22-s − 1.66·23-s + 0.392·26-s + 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s − 0.164·37-s + 0.648·38-s − 1.56·41-s − 1.82·43-s − 0.603·44-s − 1.17·46-s − 49-s + 0.277·52-s + 0.824·53-s + 0.262·58-s − 0.520·59-s − 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16650\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 37\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{16650} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 16650,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
37 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.98578431063167, −15.57852708656249, −15.25841981426829, −14.35675956384819, −13.91188435630190, −13.47124923092883, −13.02325445338722, −12.22281594895743, −11.81836502902274, −11.38808197325735, −10.39407903689146, −10.20105554648108, −9.638143590256739, −8.491136045913202, −8.177355201931188, −7.582938397098639, −6.824360543249527, −6.168576601675747, −5.603746735629769, −4.966723373178621, −4.412377765252677, −3.414170394835028, −3.072875591530518, −2.123527912338604, −1.308868393941894, 0, 1.308868393941894, 2.123527912338604, 3.072875591530518, 3.414170394835028, 4.412377765252677, 4.966723373178621, 5.603746735629769, 6.168576601675747, 6.824360543249527, 7.582938397098639, 8.177355201931188, 8.491136045913202, 9.638143590256739, 10.20105554648108, 10.39407903689146, 11.38808197325735, 11.81836502902274, 12.22281594895743, 13.02325445338722, 13.47124923092883, 13.91188435630190, 14.35675956384819, 15.25841981426829, 15.57852708656249, 15.98578431063167

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