Properties

Degree $2$
Conductor $177870$
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 + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 2·13-s − 15-s + 16-s + 4·17-s − 18-s + 20-s + 8·23-s + 24-s + 25-s + 2·26-s − 27-s + 30-s − 2·31-s − 32-s − 4·34-s + 36-s + 8·37-s + 2·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.223·20-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.182·30-s − 0.359·31-s − 0.176·32-s − 0.685·34-s + 1/6·36-s + 1.31·37-s + 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177870\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{177870} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177870,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.709580641\)
\(L(\frac12)\) \(\approx\) \(1.709580641\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
11 \( 1 \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.03309898763879, −12.75936845940198, −12.01432007492029, −11.83478134001424, −11.06243746353734, −10.91780475982662, −10.23765432165626, −9.864573284025061, −9.505342609384031, −8.831435227687869, −8.615933076882388, −7.664108165801840, −7.466795455629058, −6.967565180226378, −6.431057016560943, −5.666722771393693, −5.618290226876548, −4.870858056444430, −4.307369834080705, −3.627710744355534, −2.723277939563004, −2.632179166950287, −1.585968323637248, −1.110122961698531, −0.4955081855221097, 0.4955081855221097, 1.110122961698531, 1.585968323637248, 2.632179166950287, 2.723277939563004, 3.627710744355534, 4.307369834080705, 4.870858056444430, 5.618290226876548, 5.666722771393693, 6.431057016560943, 6.967565180226378, 7.466795455629058, 7.664108165801840, 8.615933076882388, 8.831435227687869, 9.505342609384031, 9.864573284025061, 10.23765432165626, 10.91780475982662, 11.06243746353734, 11.83478134001424, 12.01432007492029, 12.75936845940198, 13.03309898763879

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