Properties

Degree $2$
Conductor $68970$
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 + 5-s − 6-s − 4·7-s − 8-s + 9-s − 10-s + 12-s + 2·13-s + 4·14-s + 15-s + 16-s + 2·17-s − 18-s + 19-s + 20-s − 4·21-s − 8·23-s − 24-s + 25-s − 2·26-s + 27-s − 4·28-s − 6·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.872·21-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.755·28-s − 1.11·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68970\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 19\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{68970} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 68970,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 \)
19 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.43751582988310, −13.79503541893938, −13.35938712098156, −13.10090496209292, −12.29292177089829, −12.00752355455667, −11.40737969540063, −10.54493295610084, −10.08738318125064, −9.969400497393526, −9.353229854750989, −8.798817931374785, −8.469436613218283, −7.701470250106712, −7.301240854331516, −6.551664379285023, −6.276737922982503, −5.690608348154023, −5.027140022761562, −3.997332044091373, −3.448991357477112, −3.181236038738672, −2.199199826608085, −1.839575914655458, −0.8456319312248338, 0, 0.8456319312248338, 1.839575914655458, 2.199199826608085, 3.181236038738672, 3.448991357477112, 3.997332044091373, 5.027140022761562, 5.690608348154023, 6.276737922982503, 6.551664379285023, 7.301240854331516, 7.701470250106712, 8.469436613218283, 8.798817931374785, 9.353229854750989, 9.969400497393526, 10.08738318125064, 10.54493295610084, 11.40737969540063, 12.00752355455667, 12.29292177089829, 13.10090496209292, 13.35938712098156, 13.79503541893938, 14.43751582988310

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