Properties

Label 2-176610-1.1-c1-0-16
Degree $2$
Conductor $176610$
Sign $1$
Analytic cond. $1410.23$
Root an. cond. $37.5531$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 12-s + 2·13-s − 14-s + 15-s + 16-s + 6·17-s − 18-s + 4·19-s − 20-s − 21-s + 24-s + 25-s − 2·26-s − 27-s + 28-s − 30-s + 4·31-s − 32-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 − 0.288·12-s + 0.554·13-s − 0.267·14-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 − 0.218·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s − 0.182·30-s + 0.718·31-s − 0.176·32-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: $1$
Analytic conductor: \(1410.23\)
Root analytic conductor: \(37.5531\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 176610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.478450470\)
\(L(\frac12)\) \(\approx\) \(1.478450470\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
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

−13.18721016025508, −12.34486898088749, −12.11037517227634, −11.81612649078403, −11.28353798160740, −10.79314651314228, −10.27875004481591, −10.04751455817779, −9.336671496771968, −8.934172083549907, −8.295735369819998, −7.854430802647489, −7.562665318512448, −6.897838219524909, −6.509584185848202, −5.793534898160269, −5.406222370625980, −4.927091996063599, −4.203184419708981, −3.560304825612146, −3.155323039832001, −2.404944661447174, −1.494905521471232, −1.150194803546410, −0.4662814590554064, 0.4662814590554064, 1.150194803546410, 1.494905521471232, 2.404944661447174, 3.155323039832001, 3.560304825612146, 4.203184419708981, 4.927091996063599, 5.406222370625980, 5.793534898160269, 6.509584185848202, 6.897838219524909, 7.562665318512448, 7.854430802647489, 8.295735369819998, 8.934172083549907, 9.336671496771968, 10.04751455817779, 10.27875004481591, 10.79314651314228, 11.28353798160740, 11.81612649078403, 12.11037517227634, 12.34486898088749, 13.18721016025508

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