Properties

Degree $2$
Conductor $14157$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·5-s + 3·8-s − 2·10-s − 13-s − 16-s − 6·17-s + 4·19-s − 2·20-s + 8·23-s − 25-s + 26-s − 10·29-s − 5·32-s + 6·34-s + 6·37-s − 4·38-s + 6·40-s + 10·41-s − 4·43-s − 8·46-s − 8·47-s − 7·49-s + 50-s + 52-s + 10·53-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 0.632·10-s − 0.277·13-s − 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.447·20-s + 1.66·23-s − 1/5·25-s + 0.196·26-s − 1.85·29-s − 0.883·32-s + 1.02·34-s + 0.986·37-s − 0.648·38-s + 0.948·40-s + 1.56·41-s − 0.609·43-s − 1.17·46-s − 1.16·47-s − 49-s + 0.141·50-s + 0.138·52-s + 1.37·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14157 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14157 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14157\)    =    \(3^{2} \cdot 11^{2} \cdot 13\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{14157} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 14157,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.61740189388665, −16.11105347838980, −15.13215429343735, −14.85371047480668, −14.10802991940116, −13.45308754252700, −13.15175299628233, −12.81089471553639, −11.69922031278862, −11.11561211637995, −10.73938948957461, −9.816076780211433, −9.546181600582968, −9.058108176528899, −8.556640777530423, −7.615022031006164, −7.285960568066970, −6.454748414114181, −5.708475487478455, −5.072710234741976, −4.500919812548506, −3.661729626730327, −2.678489560335389, −1.890065210135159, −1.102389396031215, 0, 1.102389396031215, 1.890065210135159, 2.678489560335389, 3.661729626730327, 4.500919812548506, 5.072710234741976, 5.708475487478455, 6.454748414114181, 7.285960568066970, 7.615022031006164, 8.556640777530423, 9.058108176528899, 9.546181600582968, 9.816076780211433, 10.73938948957461, 11.11561211637995, 11.69922031278862, 12.81089471553639, 13.15175299628233, 13.45308754252700, 14.10802991940116, 14.85371047480668, 15.13215429343735, 16.11105347838980, 16.61740189388665

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