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Properties

Label 2-168e2-1.1-c1-0-126
Degree $2$
Conductor $28224$
Sign $-1$
Analytic cond. $225.369$
Root an. cond. $15.0123$
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  + 2·5-s − 2·11-s − 2·17-s − 2·23-s − 25-s + 6·29-s − 4·31-s − 6·37-s + 2·41-s − 6·53-s − 4·55-s + 12·59-s − 12·61-s + 12·67-s + 10·71-s + 12·73-s + 12·79-s + 12·83-s − 4·85-s − 14·89-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 4·115-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.603·11-s − 0.485·17-s − 0.417·23-s − 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.986·37-s + 0.312·41-s − 0.824·53-s − 0.539·55-s + 1.56·59-s − 1.53·61-s + 1.46·67-s + 1.18·71-s + 1.40·73-s + 1.35·79-s + 1.31·83-s − 0.433·85-s − 1.48·89-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.373·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(225.369\)
Root analytic conductor: \(15.0123\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 28224,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−15.49728545016949, −15.02534207152513, −14.14762733862665, −13.95645897561193, −13.47176120914847, −12.75953593680346, −12.45603529036716, −11.75615314246829, −11.03549367423849, −10.64935449843986, −10.02956132839025, −9.564411167676348, −9.046886513437834, −8.297569709882017, −7.912937612969561, −7.083945070287202, −6.518688622454353, −6.023463160281178, −5.248158910299300, −4.952074548269805, −4.004183412366822, −3.371195476963263, −2.411091682289289, −2.077289354049083, −1.102439462141840, 0, 1.102439462141840, 2.077289354049083, 2.411091682289289, 3.371195476963263, 4.004183412366822, 4.952074548269805, 5.248158910299300, 6.023463160281178, 6.518688622454353, 7.083945070287202, 7.912937612969561, 8.297569709882017, 9.046886513437834, 9.564411167676348, 10.02956132839025, 10.64935449843986, 11.03549367423849, 11.75615314246829, 12.45603529036716, 12.75953593680346, 13.47176120914847, 13.95645897561193, 14.14762733862665, 15.02534207152513, 15.49728545016949

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