Properties

Degree $2$
Conductor $227850$
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 − 4·11-s + 12-s + 6·13-s + 16-s + 2·17-s − 18-s − 4·19-s + 4·22-s + 8·23-s − 24-s − 6·26-s + 27-s + 6·29-s + 31-s − 32-s − 4·33-s − 2·34-s + 36-s + 2·37-s + 4·38-s + 6·39-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 − 1.20·11-s + 0.288·12-s + 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.852·22-s + 1.66·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s + 1.11·29-s + 0.179·31-s − 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s + 0.328·37-s + 0.648·38-s + 0.960·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 227850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(227850\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 31\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{227850} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 227850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.364931620\)
\(L(\frac12)\) \(\approx\) \(3.364931620\)
\(L(\frac{3}{2})\) 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 \( 1 \)
31 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.05695143180753, −12.63190615797491, −11.82544846562565, −11.54750973671806, −10.81598555776502, −10.55953394168378, −10.26158499895667, −9.664470136224583, −8.991287065593408, −8.664090997565254, −8.344707700578887, −7.925737731990788, −7.344688184044095, −6.752945504647617, −6.473833321498817, −5.733817763801822, −5.284661194280694, −4.676090217567227, −4.021439720168075, −3.324966426206031, −3.063992272977788, −2.312341059698464, −1.878084337323653, −0.9202774471877364, −0.6988901306599072, 0.6988901306599072, 0.9202774471877364, 1.878084337323653, 2.312341059698464, 3.063992272977788, 3.324966426206031, 4.021439720168075, 4.676090217567227, 5.284661194280694, 5.733817763801822, 6.473833321498817, 6.752945504647617, 7.344688184044095, 7.925737731990788, 8.344707700578887, 8.664090997565254, 8.991287065593408, 9.664470136224583, 10.26158499895667, 10.55953394168378, 10.81598555776502, 11.54750973671806, 11.82544846562565, 12.63190615797491, 13.05695143180753

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