Properties

Label 2-87120-1.1-c1-0-100
Degree $2$
Conductor $87120$
Sign $-1$
Analytic cond. $695.656$
Root an. cond. $26.3753$
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  − 5-s − 3·7-s + 2·13-s + 6·17-s − 5·19-s + 4·23-s + 25-s + 6·29-s + 5·31-s + 3·35-s − 11·37-s + 4·41-s − 4·43-s − 6·47-s + 2·49-s + 8·53-s − 2·59-s − 61-s − 2·65-s + 7·67-s − 10·71-s + 11·73-s + 79-s − 10·83-s − 6·85-s + 14·89-s − 6·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s + 0.554·13-s + 1.45·17-s − 1.14·19-s + 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.898·31-s + 0.507·35-s − 1.80·37-s + 0.624·41-s − 0.609·43-s − 0.875·47-s + 2/7·49-s + 1.09·53-s − 0.260·59-s − 0.128·61-s − 0.248·65-s + 0.855·67-s − 1.18·71-s + 1.28·73-s + 0.112·79-s − 1.09·83-s − 0.650·85-s + 1.48·89-s − 0.628·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87120\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(695.656\)
Root analytic conductor: \(26.3753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{87120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 87120,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good7 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−14.17959280924375, −13.58096252475258, −13.13829885711855, −12.59396116211993, −12.27545463379364, −11.78626917829898, −11.19026977189782, −10.53422465213818, −10.20750376329346, −9.792186261029419, −9.072728579110026, −8.586634442439847, −8.219352248567085, −7.539405710688204, −6.954396238371669, −6.428605143476097, −6.160398917352962, −5.277419987986288, −4.898328398664439, −4.027547531267561, −3.596657299504207, −3.058652062582361, −2.540797776700670, −1.510468334649956, −0.8388712340665945, 0, 0.8388712340665945, 1.510468334649956, 2.540797776700670, 3.058652062582361, 3.596657299504207, 4.027547531267561, 4.898328398664439, 5.277419987986288, 6.160398917352962, 6.428605143476097, 6.954396238371669, 7.539405710688204, 8.219352248567085, 8.586634442439847, 9.072728579110026, 9.792186261029419, 10.20750376329346, 10.53422465213818, 11.19026977189782, 11.78626917829898, 12.27545463379364, 12.59396116211993, 13.13829885711855, 13.58096252475258, 14.17959280924375

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