Properties

Label 2-179520-1.1-c1-0-177
Degree $2$
Conductor $179520$
Sign $-1$
Analytic cond. $1433.47$
Root an. cond. $37.8612$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 2·7-s + 9-s + 11-s − 15-s + 17-s − 2·21-s + 4·23-s + 25-s − 27-s − 2·29-s − 4·31-s − 33-s + 2·35-s + 2·37-s − 6·43-s + 45-s − 3·49-s − 51-s + 6·53-s + 55-s + 14·59-s + 2·61-s + 2·63-s + 14·67-s − 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.258·15-s + 0.242·17-s − 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.174·33-s + 0.338·35-s + 0.328·37-s − 0.914·43-s + 0.149·45-s − 3/7·49-s − 0.140·51-s + 0.824·53-s + 0.134·55-s + 1.82·59-s + 0.256·61-s + 0.251·63-s + 1.71·67-s − 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(179520\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(1433.47\)
Root analytic conductor: \(37.8612\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{179520} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 179520,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−13.30187977699424, −12.91058519066106, −12.50073111448552, −11.91866613149043, −11.39710780584658, −11.12374186841094, −10.74147849539344, −9.924704512560677, −9.809873420612528, −9.182273516265264, −8.507956879364154, −8.277017409783224, −7.548816484069784, −6.987487406919206, −6.720462820090427, −6.027702658347072, −5.436734558243671, −5.175944221016305, −4.665531901041093, −3.885385104819721, −3.570727099398863, −2.613537228078720, −2.140326818001875, −1.378924844273427, −0.9753148494874213, 0, 0.9753148494874213, 1.378924844273427, 2.140326818001875, 2.613537228078720, 3.570727099398863, 3.885385104819721, 4.665531901041093, 5.175944221016305, 5.436734558243671, 6.027702658347072, 6.720462820090427, 6.987487406919206, 7.548816484069784, 8.277017409783224, 8.507956879364154, 9.182273516265264, 9.809873420612528, 9.924704512560677, 10.74147849539344, 11.12374186841094, 11.39710780584658, 11.91866613149043, 12.50073111448552, 12.91058519066106, 13.30187977699424

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