Properties

Label 2-185020-1.1-c1-0-17
Degree $2$
Conductor $185020$
Sign $1$
Analytic cond. $1477.39$
Root an. cond. $38.4368$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5-s + 2·7-s + 9-s − 11-s − 2·13-s − 2·15-s − 6·17-s − 8·19-s + 4·21-s − 6·23-s + 25-s − 4·27-s − 2·33-s − 2·35-s − 2·37-s − 4·39-s − 6·41-s + 6·43-s − 45-s − 2·47-s − 3·49-s − 12·51-s − 6·53-s + 55-s − 16·57-s + 4·59-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s − 1.45·17-s − 1.83·19-s + 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 0.348·33-s − 0.338·35-s − 0.328·37-s − 0.640·39-s − 0.937·41-s + 0.914·43-s − 0.149·45-s − 0.291·47-s − 3/7·49-s − 1.68·51-s − 0.824·53-s + 0.134·55-s − 2.11·57-s + 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185020\)    =    \(2^{2} \cdot 5 \cdot 11 \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(1477.39\)
Root analytic conductor: \(38.4368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{185020} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 185020,\ (\ :1/2),\ 1)\)

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 \( 1 \)
5 \( 1 + T \)
11 \( 1 + T \)
29 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.79656996251852, −13.02980536736344, −12.88403038975875, −12.23356700179523, −11.70170290326615, −11.22697818401170, −10.80710611156601, −10.31135393558898, −9.784211269978785, −9.150438866298733, −8.733546158287765, −8.286300817772040, −8.134398830689217, −7.432385496667205, −7.078248923372111, −6.315571065389770, −5.970356093296208, −5.110677425968878, −4.453159760069358, −4.347899981458123, −3.652057534231570, −3.007435374823227, −2.337005968018413, −2.068759776475011, −1.481318870665060, 0, 0, 1.481318870665060, 2.068759776475011, 2.337005968018413, 3.007435374823227, 3.652057534231570, 4.347899981458123, 4.453159760069358, 5.110677425968878, 5.970356093296208, 6.315571065389770, 7.078248923372111, 7.432385496667205, 8.134398830689217, 8.286300817772040, 8.733546158287765, 9.150438866298733, 9.784211269978785, 10.31135393558898, 10.80710611156601, 11.22697818401170, 11.70170290326615, 12.23356700179523, 12.88403038975875, 13.02980536736344, 13.79656996251852

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