Properties

Degree $2$
Conductor $127296$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4·11-s − 13-s − 17-s + 4·19-s − 25-s − 2·29-s − 8·31-s + 2·37-s − 2·41-s + 4·43-s − 8·47-s − 7·49-s − 10·53-s − 8·55-s + 4·59-s − 14·61-s + 2·65-s + 4·67-s − 14·73-s − 8·79-s − 4·83-s + 2·85-s + 6·89-s − 8·95-s − 6·97-s + 101-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.20·11-s − 0.277·13-s − 0.242·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s − 49-s − 1.37·53-s − 1.07·55-s + 0.520·59-s − 1.79·61-s + 0.248·65-s + 0.488·67-s − 1.63·73-s − 0.900·79-s − 0.439·83-s + 0.216·85-s + 0.635·89-s − 0.820·95-s − 0.609·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(127296\)    =    \(2^{6} \cdot 3^{2} \cdot 13 \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{127296} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 127296,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8090075530\)
\(L(\frac12)\) \(\approx\) \(0.8090075530\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + T \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.43713563256894, −12.99721261207346, −12.48376563423399, −11.91961575193228, −11.69171888277878, −11.04324012708031, −10.94646535819905, −9.887834831594222, −9.680994942107313, −9.095986325279788, −8.680603268462558, −8.013292094390448, −7.525459647806031, −7.232133818102772, −6.549232252282178, −6.086172068659097, −5.456255454454300, −4.797565249231709, −4.327803268205541, −3.712328480887103, −3.355955832233928, −2.689389942449642, −1.689672098247563, −1.362382511101892, −0.2743457357220000, 0.2743457357220000, 1.362382511101892, 1.689672098247563, 2.689389942449642, 3.355955832233928, 3.712328480887103, 4.327803268205541, 4.797565249231709, 5.456255454454300, 6.086172068659097, 6.549232252282178, 7.232133818102772, 7.525459647806031, 8.013292094390448, 8.680603268462558, 9.095986325279788, 9.680994942107313, 9.887834831594222, 10.94646535819905, 11.04324012708031, 11.69171888277878, 11.91961575193228, 12.48376563423399, 12.99721261207346, 13.43713563256894

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