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 − 6·17-s − 18-s − 4·19-s + 20-s + 24-s + 25-s − 2·26-s − 27-s + 6·29-s + 30-s + 4·31-s − 32-s + 6·34-s + 36-s + 2·37-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 − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 1.11·29-s + 0.182·30-s + 0.718·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-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

\(L(1)\) \(\approx\) \(1.814639715\)
\(L(\frac12)\) \(\approx\) \(1.814639715\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.19785685999110, −12.53034819841689, −12.27566800476913, −11.61805796643317, −11.11691432183424, −10.84858405659929, −10.35305841684308, −9.933160395717265, −9.354301572852727, −8.898987880788033, −8.381308980968038, −8.118594898441874, −7.275920317208257, −6.755561955719559, −6.421587009191095, −6.105594881099196, −5.306072769112325, −4.932753947885352, −4.094997390262839, −3.886075241072965, −2.790012768065890, −2.338842525453048, −1.863774568724698, −0.9182040863911166, −0.5636024151750719, 0.5636024151750719, 0.9182040863911166, 1.863774568724698, 2.338842525453048, 2.790012768065890, 3.886075241072965, 4.094997390262839, 4.932753947885352, 5.306072769112325, 6.105594881099196, 6.421587009191095, 6.755561955719559, 7.275920317208257, 8.118594898441874, 8.381308980968038, 8.898987880788033, 9.354301572852727, 9.933160395717265, 10.35305841684308, 10.84858405659929, 11.11691432183424, 11.61805796643317, 12.27566800476913, 12.53034819841689, 13.19785685999110

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