Properties

Label 2-168e2-1.1-c1-0-124
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  + 2·11-s − 2·13-s + 4·17-s + 4·19-s − 6·23-s − 5·25-s − 2·29-s + 6·37-s + 8·41-s − 8·43-s + 4·47-s − 6·53-s − 14·61-s + 4·67-s − 2·71-s + 2·73-s − 4·79-s + 12·83-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.603·11-s − 0.554·13-s + 0.970·17-s + 0.917·19-s − 1.25·23-s − 25-s − 0.371·29-s + 0.986·37-s + 1.24·41-s − 1.21·43-s + 0.583·47-s − 0.824·53-s − 1.79·61-s + 0.488·67-s − 0.237·71-s + 0.234·73-s − 0.450·79-s + 1.31·83-s − 0.609·97-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.53913751299979, −14.78565301501815, −14.50183918970803, −13.83285189433762, −13.55693868680916, −12.70511777651946, −12.15948160615268, −11.86376663730697, −11.28594926671435, −10.60571325690179, −9.937601001411790, −9.514854739013179, −9.190448556013459, −8.105239798876311, −7.861801141925867, −7.309508644016100, −6.531133825768713, −5.900014914382841, −5.487177283374321, −4.651900155410895, −4.026465140240887, −3.417753755010638, −2.670986669948203, −1.833054125399473, −1.094598920755460, 0, 1.094598920755460, 1.833054125399473, 2.670986669948203, 3.417753755010638, 4.026465140240887, 4.651900155410895, 5.487177283374321, 5.900014914382841, 6.531133825768713, 7.309508644016100, 7.861801141925867, 8.105239798876311, 9.190448556013459, 9.514854739013179, 9.937601001411790, 10.60571325690179, 11.28594926671435, 11.86376663730697, 12.15948160615268, 12.70511777651946, 13.55693868680916, 13.83285189433762, 14.50183918970803, 14.78565301501815, 15.53913751299979

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