Properties

Label 2-168e2-1.1-c1-0-112
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  + 2·13-s − 8·19-s − 5·25-s − 4·31-s + 10·37-s + 8·43-s + 14·61-s − 16·67-s + 10·73-s + 4·79-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.554·13-s − 1.83·19-s − 25-s − 0.718·31-s + 1.64·37-s + 1.21·43-s + 1.79·61-s − 1.95·67-s + 1.17·73-s + 0.450·79-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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: $\chi_{28224} (1, \cdot )$
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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.46821528320345, −14.91173069378384, −14.50329902722202, −13.90411500315802, −13.20794468560532, −12.92674105639394, −12.37602627202987, −11.63357499264642, −11.19733968919178, −10.63601645062315, −10.19192470965491, −9.366668611309472, −9.052111428031050, −8.250002425519016, −7.931848058418903, −7.171098751085239, −6.504936316197059, −5.987628313104290, −5.503271996771148, −4.511139990908336, −4.124621445637085, −3.478589533669437, −2.493298667434967, −2.010390926931963, −1.015397726855132, 0, 1.015397726855132, 2.010390926931963, 2.493298667434967, 3.478589533669437, 4.124621445637085, 4.511139990908336, 5.503271996771148, 5.987628313104290, 6.504936316197059, 7.171098751085239, 7.931848058418903, 8.250002425519016, 9.052111428031050, 9.366668611309472, 10.19192470965491, 10.63601645062315, 11.19733968919178, 11.63357499264642, 12.37602627202987, 12.92674105639394, 13.20794468560532, 13.90411500315802, 14.50329902722202, 14.91173069378384, 15.46821528320345

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