Properties

Label 2-388080-1.1-c1-0-102
Degree $2$
Conductor $388080$
Sign $-1$
Analytic cond. $3098.83$
Root an. cond. $55.6671$
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 − 11-s − 6·13-s − 5·17-s − 5·19-s + 23-s + 25-s − 3·29-s − 6·37-s + 4·41-s − 11·43-s + 4·47-s − 53-s + 55-s − 15·59-s + 15·61-s + 6·65-s + 10·67-s + 12·71-s − 4·73-s + 2·79-s − 15·83-s + 5·85-s − 7·89-s + 5·95-s − 5·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s − 1.66·13-s − 1.21·17-s − 1.14·19-s + 0.208·23-s + 1/5·25-s − 0.557·29-s − 0.986·37-s + 0.624·41-s − 1.67·43-s + 0.583·47-s − 0.137·53-s + 0.134·55-s − 1.95·59-s + 1.92·61-s + 0.744·65-s + 1.22·67-s + 1.42·71-s − 0.468·73-s + 0.225·79-s − 1.64·83-s + 0.542·85-s − 0.741·89-s + 0.512·95-s − 0.507·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(388080\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(3098.83\)
Root analytic conductor: \(55.6671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 388080,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 15 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−12.68112022200786, −12.34930654796403, −11.71947045973199, −11.34640423529443, −10.95670470882705, −10.35370310215283, −10.11824546587686, −9.464698447264176, −9.069719209055405, −8.618201610114490, −8.036068486262485, −7.781075297110306, −7.029610285683128, −6.819122382204345, −6.424735765103511, −5.579262524352183, −5.156671664714450, −4.770630906167715, −4.169066561797941, −3.857041344402898, −3.040669088549314, −2.545768324138650, −2.104738562227933, −1.537474741567472, −0.4574138864654198, 0, 0.4574138864654198, 1.537474741567472, 2.104738562227933, 2.545768324138650, 3.040669088549314, 3.857041344402898, 4.169066561797941, 4.770630906167715, 5.156671664714450, 5.579262524352183, 6.424735765103511, 6.819122382204345, 7.029610285683128, 7.781075297110306, 8.036068486262485, 8.618201610114490, 9.069719209055405, 9.464698447264176, 10.11824546587686, 10.35370310215283, 10.95670470882705, 11.34640423529443, 11.71947045973199, 12.34930654796403, 12.68112022200786

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