Properties

Degree $2$
Conductor $53312$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s + 9-s + 6·11-s + 2·13-s + 4·15-s − 17-s + 6·23-s − 25-s + 4·27-s + 10·29-s − 2·31-s − 12·33-s − 6·37-s − 4·39-s + 6·41-s + 8·43-s − 2·45-s + 2·51-s + 10·53-s − 12·55-s − 8·59-s + 14·61-s − 4·65-s − 4·67-s − 12·69-s + 2·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 1.03·15-s − 0.242·17-s + 1.25·23-s − 1/5·25-s + 0.769·27-s + 1.85·29-s − 0.359·31-s − 2.08·33-s − 0.986·37-s − 0.640·39-s + 0.937·41-s + 1.21·43-s − 0.298·45-s + 0.280·51-s + 1.37·53-s − 1.61·55-s − 1.04·59-s + 1.79·61-s − 0.496·65-s − 0.488·67-s − 1.44·69-s + 0.237·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53312\)    =    \(2^{6} \cdot 7^{2} \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{53312} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 53312,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.856430508\)
\(L(\frac12)\) \(\approx\) \(1.856430508\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
17 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.39691649175945, −14.05419168816980, −13.42380123636636, −12.62942712895126, −12.24441076413505, −11.85283245313355, −11.44151245273949, −10.96659495147294, −10.62475748121399, −9.854959156986091, −9.169876698798975, −8.711339148361564, −8.316168331670746, −7.385493800114174, −6.962635747033667, −6.458693135156712, −6.019834077135944, −5.354253926211986, −4.651890939928081, −4.203161025762754, −3.628246456680653, −2.982220538694515, −1.948162313686625, −0.8632357709008840, −0.7569036862653294, 0.7569036862653294, 0.8632357709008840, 1.948162313686625, 2.982220538694515, 3.628246456680653, 4.203161025762754, 4.651890939928081, 5.354253926211986, 6.019834077135944, 6.458693135156712, 6.962635747033667, 7.385493800114174, 8.316168331670746, 8.711339148361564, 9.169876698798975, 9.854959156986091, 10.62475748121399, 10.96659495147294, 11.44151245273949, 11.85283245313355, 12.24441076413505, 12.62942712895126, 13.42380123636636, 14.05419168816980, 14.39691649175945

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