Properties

Degree $2$
Conductor $79350$
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 + 2·13-s + 16-s − 6·17-s − 18-s − 4·19-s + 4·22-s + 24-s − 2·26-s − 27-s − 2·29-s − 32-s + 4·33-s + 6·34-s + 36-s − 2·37-s + 4·38-s − 2·39-s + 10·41-s − 4·43-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 + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.852·22-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.371·29-s − 0.176·32-s + 0.696·33-s + 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.648·38-s − 0.320·39-s + 1.56·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(79350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{79350} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 79350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1953882002\)
\(L(\frac12)\) \(\approx\) \(0.1953882002\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.85599311073112, −13.35165805986912, −12.97514392006617, −12.57333538086584, −11.89163092318856, −11.25106595600727, −11.02352624013127, −10.54797512865229, −10.11160299118754, −9.508476016724468, −8.851905012670668, −8.509487506289122, −7.973252342277131, −7.285622889277038, −6.945623394884020, −6.189311592330829, −5.913120740881480, −5.206652737219544, −4.536259543857528, −4.093281158317842, −3.193920890771116, −2.491740781822879, −1.984149525173015, −1.191459177666320, −0.1743130924412017, 0.1743130924412017, 1.191459177666320, 1.984149525173015, 2.491740781822879, 3.193920890771116, 4.093281158317842, 4.536259543857528, 5.206652737219544, 5.913120740881480, 6.189311592330829, 6.945623394884020, 7.285622889277038, 7.973252342277131, 8.509487506289122, 8.851905012670668, 9.508476016724468, 10.11160299118754, 10.54797512865229, 11.02352624013127, 11.25106595600727, 11.89163092318856, 12.57333538086584, 12.97514392006617, 13.35165805986912, 13.85599311073112

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