Properties

Degree $2$
Conductor $7350$
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 − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s − 7·13-s + 16-s + 4·17-s − 18-s + 19-s + 22-s − 23-s + 24-s + 7·26-s − 27-s − 8·29-s + 6·31-s − 32-s + 33-s − 4·34-s + 36-s + 3·37-s − 38-s + 7·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.94·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.229·19-s + 0.213·22-s − 0.208·23-s + 0.204·24-s + 1.37·26-s − 0.192·27-s − 1.48·29-s + 1.07·31-s − 0.176·32-s + 0.174·33-s − 0.685·34-s + 1/6·36-s + 0.493·37-s − 0.162·38-s + 1.12·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{7350} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7350,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−17.44928731148810, −16.77963262758127, −16.52697130679752, −15.79135025554324, −15.10353150005829, −14.62202894106123, −14.01665750611629, −13.01819689926425, −12.54623440680959, −11.90250925556790, −11.52283080269226, −10.64967535860848, −10.13576883032906, −9.617401549009076, −9.100581750463638, −8.037634029136000, −7.490820353257988, −7.178598498797547, −6.127434600604596, −5.559292555125702, −4.859066004863428, −4.012647805786473, −2.864623099255355, −2.214713427086070, −1.034465352669726, 0, 1.034465352669726, 2.214713427086070, 2.864623099255355, 4.012647805786473, 4.859066004863428, 5.559292555125702, 6.127434600604596, 7.178598498797547, 7.490820353257988, 8.037634029136000, 9.100581750463638, 9.617401549009076, 10.13576883032906, 10.64967535860848, 11.52283080269226, 11.90250925556790, 12.54623440680959, 13.01819689926425, 14.01665750611629, 14.62202894106123, 15.10353150005829, 15.79135025554324, 16.52697130679752, 16.77963262758127, 17.44928731148810

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