Properties

Degree $2$
Conductor $7350$
Sign $1$
Motivic weight $1$
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 − 6-s − 8-s + 9-s − 5·11-s + 12-s + 5·13-s + 16-s + 4·17-s − 18-s − 7·19-s + 5·22-s − 23-s − 24-s − 5·26-s + 27-s − 2·31-s − 32-s − 5·33-s − 4·34-s + 36-s − 37-s + 7·38-s + 5·39-s + 5·41-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 − 1.50·11-s + 0.288·12-s + 1.38·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 1.60·19-s + 1.06·22-s − 0.208·23-s − 0.204·24-s − 0.980·26-s + 0.192·27-s − 0.359·31-s − 0.176·32-s − 0.870·33-s − 0.685·34-s + 1/6·36-s − 0.164·37-s + 1.13·38-s + 0.800·39-s + 0.780·41-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: \(0\)
Selberg data: \((2,\ 7350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.584718749\)
\(L(\frac12)\) \(\approx\) \(1.584718749\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−17.08000156108227, −16.50568830466587, −15.91632013262838, −15.49412453627855, −14.87157684141088, −14.28390583976288, −13.39112862123888, −13.10228335871040, −12.43665528701020, −11.62747025934921, −10.85931020216942, −10.39893592808680, −10.00330858956444, −9.001925675878700, −8.529759885581027, −8.061335777140359, −7.452094034506997, −6.648662536199093, −5.886888902019116, −5.241107351049026, −4.126323779565532, −3.432743983543459, −2.546527963260244, −1.844466706480648, −0.6769932695286660, 0.6769932695286660, 1.844466706480648, 2.546527963260244, 3.432743983543459, 4.126323779565532, 5.241107351049026, 5.886888902019116, 6.648662536199093, 7.452094034506997, 8.061335777140359, 8.529759885581027, 9.001925675878700, 10.00330858956444, 10.39893592808680, 10.85931020216942, 11.62747025934921, 12.43665528701020, 13.10228335871040, 13.39112862123888, 14.28390583976288, 14.87157684141088, 15.49412453627855, 15.91632013262838, 16.50568830466587, 17.08000156108227

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