Properties

Label 2-168e2-1.1-c1-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 6·11-s − 6·13-s + 2·17-s + 4·19-s + 2·23-s − 25-s − 8·29-s − 4·31-s + 6·37-s − 10·41-s + 4·43-s + 4·47-s + 4·53-s − 12·55-s − 12·59-s − 2·61-s − 12·65-s − 12·67-s + 6·71-s + 2·73-s − 8·79-s + 4·85-s + 14·89-s + 8·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.80·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s + 0.417·23-s − 1/5·25-s − 1.48·29-s − 0.718·31-s + 0.986·37-s − 1.56·41-s + 0.609·43-s + 0.583·47-s + 0.549·53-s − 1.61·55-s − 1.56·59-s − 0.256·61-s − 1.48·65-s − 1.46·67-s + 0.712·71-s + 0.234·73-s − 0.900·79-s + 0.433·85-s + 1.48·89-s + 0.820·95-s + 0.203·97-s + 0.0995·101-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: \(0\)
Selberg data: \((2,\ 28224,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.340276107\)
\(L(\frac12)\) \(\approx\) \(1.340276107\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.07501456997743, −14.80732197654590, −14.08895761384143, −13.49985299940254, −13.23520796262457, −12.53717554325225, −12.19528733824409, −11.40424451908074, −10.82633881006276, −10.21188802430051, −9.832917432211684, −9.415755257713374, −8.758849395598375, −7.821776343201235, −7.476634067463453, −7.175221952350288, −5.988856869116287, −5.676896061232039, −5.046708964115618, −4.711064792332173, −3.540953280957518, −2.866722517258167, −2.304253481051797, −1.678260613403831, −0.4217586548308436, 0.4217586548308436, 1.678260613403831, 2.304253481051797, 2.866722517258167, 3.540953280957518, 4.711064792332173, 5.046708964115618, 5.676896061232039, 5.988856869116287, 7.175221952350288, 7.476634067463453, 7.821776343201235, 8.758849395598375, 9.415755257713374, 9.832917432211684, 10.21188802430051, 10.82633881006276, 11.40424451908074, 12.19528733824409, 12.53717554325225, 13.23520796262457, 13.49985299940254, 14.08895761384143, 14.80732197654590, 15.07501456997743

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