Properties

Label 2-11424-1.1-c1-0-1
Degree $2$
Conductor $11424$
Sign $1$
Analytic cond. $91.2210$
Root an. cond. $9.55097$
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  − 3-s + 2·5-s + 7-s + 9-s − 4·11-s − 2·13-s − 2·15-s + 17-s − 8·19-s − 21-s − 25-s − 27-s + 2·29-s − 8·31-s + 4·33-s + 2·35-s + 2·37-s + 2·39-s − 6·41-s + 8·43-s + 2·45-s − 8·47-s + 49-s − 51-s + 6·53-s − 8·55-s + 8·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s + 0.242·17-s − 1.83·19-s − 0.218·21-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.338·35-s + 0.328·37-s + 0.320·39-s − 0.937·41-s + 1.21·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s − 0.140·51-s + 0.824·53-s − 1.07·55-s + 1.05·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11424\)    =    \(2^{5} \cdot 3 \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(91.2210\)
Root analytic conductor: \(9.55097\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 11424,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.330564220\)
\(L(\frac12)\) \(\approx\) \(1.330564220\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−16.62159792228843, −15.97109115587702, −15.21333635542709, −14.81836617197687, −14.19915891095805, −13.45048906543703, −12.99801513754506, −12.55822802206837, −11.89037442813184, −11.06982185763235, −10.65164902744626, −10.15826910587859, −9.589302739708166, −8.815799284730997, −8.153844676967991, −7.507857101270178, −6.810980783790409, −6.062948280479294, −5.579974105704396, −4.939632289743711, −4.350225957410654, −3.317824175263968, −2.189000811640283, −1.957740457476233, −0.5205477587777299, 0.5205477587777299, 1.957740457476233, 2.189000811640283, 3.317824175263968, 4.350225957410654, 4.939632289743711, 5.579974105704396, 6.062948280479294, 6.810980783790409, 7.507857101270178, 8.153844676967991, 8.815799284730997, 9.589302739708166, 10.15826910587859, 10.65164902744626, 11.06982185763235, 11.89037442813184, 12.55822802206837, 12.99801513754506, 13.45048906543703, 14.19915891095805, 14.81836617197687, 15.21333635542709, 15.97109115587702, 16.62159792228843

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