Properties

Label 2-388080-1.1-c1-0-134
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 11-s + 6·13-s − 3·17-s + 3·19-s − 23-s + 25-s + 7·29-s + 4·31-s − 6·37-s − 4·41-s + 9·43-s − 3·53-s + 55-s − 5·59-s − 5·61-s + 6·65-s + 2·67-s − 4·71-s + 16·73-s − 2·79-s + 3·83-s − 3·85-s − 9·89-s + 3·95-s + 7·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.301·11-s + 1.66·13-s − 0.727·17-s + 0.688·19-s − 0.208·23-s + 1/5·25-s + 1.29·29-s + 0.718·31-s − 0.986·37-s − 0.624·41-s + 1.37·43-s − 0.412·53-s + 0.134·55-s − 0.650·59-s − 0.640·61-s + 0.744·65-s + 0.244·67-s − 0.474·71-s + 1.87·73-s − 0.225·79-s + 0.329·83-s − 0.325·85-s − 0.953·89-s + 0.307·95-s + 0.710·97-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: \(0\)
Selberg data: \((2,\ 388080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.895896651\)
\(L(\frac12)\) \(\approx\) \(3.895896651\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−12.48410146658104, −12.02210902614880, −11.50440448028561, −11.07062976077268, −10.71761490223622, −10.21209192615991, −9.803232417286879, −9.185585280110481, −8.855372192789121, −8.448498799008846, −8.001852937632687, −7.412703141435613, −6.789496563271444, −6.424974881248962, −6.071282575424060, −5.573614111809187, −4.926628407548467, −4.522611229107841, −3.897028309141494, −3.433591219603546, −2.903353585217638, −2.295104261409240, −1.607820485836579, −1.156811422376324, −0.5407867080018531, 0.5407867080018531, 1.156811422376324, 1.607820485836579, 2.295104261409240, 2.903353585217638, 3.433591219603546, 3.897028309141494, 4.522611229107841, 4.926628407548467, 5.573614111809187, 6.071282575424060, 6.424974881248962, 6.789496563271444, 7.412703141435613, 8.001852937632687, 8.448498799008846, 8.855372192789121, 9.185585280110481, 9.803232417286879, 10.21209192615991, 10.71761490223622, 11.07062976077268, 11.50440448028561, 12.02210902614880, 12.48410146658104

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