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

\(L(1)\) \(\approx\) \(0.4321388969\)
\(L(\frac12)\) \(\approx\) \(0.4321388969\)
\(L(\frac{3}{2})\) 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

−16.97824210131486, −16.77990441421277, −15.89627162224434, −15.57370451745024, −15.05091845308730, −14.17000948119554, −13.46382180083279, −12.90260088626110, −12.21904153225195, −11.60168988360637, −11.17385609752259, −10.27748634177525, −9.958741059397583, −9.452137972373674, −8.377147165956598, −7.985633955569126, −7.090860945971832, −6.873238814987808, −5.727490183959086, −5.145040815186216, −4.642020551879339, −3.304368930041897, −2.581048920533366, −1.708285115717783, −0.3659856104643310, 0.3659856104643310, 1.708285115717783, 2.581048920533366, 3.304368930041897, 4.642020551879339, 5.145040815186216, 5.727490183959086, 6.873238814987808, 7.090860945971832, 7.985633955569126, 8.377147165956598, 9.452137972373674, 9.958741059397583, 10.27748634177525, 11.17385609752259, 11.60168988360637, 12.21904153225195, 12.90260088626110, 13.46382180083279, 14.17000948119554, 15.05091845308730, 15.57370451745024, 15.89627162224434, 16.77990441421277, 16.97824210131486

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