Properties

Degree $2$
Conductor $176610$
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 + 7-s − 8-s + 9-s − 10-s − 4·11-s + 12-s − 2·13-s − 14-s + 15-s + 16-s − 2·17-s − 18-s + 4·19-s + 20-s + 21-s + 4·22-s − 8·23-s − 24-s + 25-s + 2·26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s + 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176610\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 29^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{176610} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 176610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.491856788\)
\(L(\frac12)\) \(\approx\) \(2.491856788\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
29 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.32328366009220, −12.57869204939037, −12.25046135723730, −11.82783729861671, −11.05280873631490, −10.80674744773976, −10.11941612429238, −9.865740173614149, −9.484939033730103, −8.864408788545646, −8.308341524667870, −7.951291833531719, −7.601466525092921, −7.051231056371469, −6.457986534815416, −5.876646281230090, −5.318352237691818, −4.899311715978071, −4.111652974496801, −3.663318802286009, −2.606697003828880, −2.513930515053446, −2.045913530159428, −1.099123629408161, −0.5287568742811910, 0.5287568742811910, 1.099123629408161, 2.045913530159428, 2.513930515053446, 2.606697003828880, 3.663318802286009, 4.111652974496801, 4.899311715978071, 5.318352237691818, 5.876646281230090, 6.457986534815416, 7.051231056371469, 7.601466525092921, 7.951291833531719, 8.308341524667870, 8.864408788545646, 9.484939033730103, 9.865740173614149, 10.11941612429238, 10.80674744773976, 11.05280873631490, 11.82783729861671, 12.25046135723730, 12.57869204939037, 13.32328366009220

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