Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·9-s − 4·16-s − 9·25-s − 5·27-s − 14·31-s − 6·37-s − 4·48-s + 10·49-s + 14·67-s − 9·75-s + 81-s − 14·93-s − 14·97-s − 32·103-s − 6·111-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s − 16-s − 9/5·25-s − 0.962·27-s − 2.51·31-s − 0.986·37-s − 0.577·48-s + 10/7·49-s + 1.71·67-s − 1.03·75-s + 1/9·81-s − 1.45·93-s − 1.42·97-s − 3.15·103-s − 0.569·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(131769\)    =    \(3^{2} \cdot 11^{4}\)
Sign: $-1$
Analytic conductor: \(8.40170\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 131769,\ (\ :1/2, 1/2),\ -1)\)

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$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad3$C_2$ \( 1 - T + p T^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.a_bt
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.31.o_eh
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.59.a_dp
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.67.ao_hb
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.71.a_fd
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.a_abv
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.97.o_jj
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.217310407186341395835167909208, −8.705384628331600900155507087465, −8.151019817538610002230465773225, −7.79650621472104891518165412451, −7.12187588616847621428342169043, −6.84640834678184498853329498743, −6.04331122529311853759013122154, −5.45433923545548060691478733761, −5.27344529292695596967361046797, −4.10352174575273595075153470227, −3.91131207505118194555587511681, −3.14418397915721421083734845455, −2.31417648620544719198281734415, −1.82213114349074882841569099090, 0, 1.82213114349074882841569099090, 2.31417648620544719198281734415, 3.14418397915721421083734845455, 3.91131207505118194555587511681, 4.10352174575273595075153470227, 5.27344529292695596967361046797, 5.45433923545548060691478733761, 6.04331122529311853759013122154, 6.84640834678184498853329498743, 7.12187588616847621428342169043, 7.79650621472104891518165412451, 8.151019817538610002230465773225, 8.705384628331600900155507087465, 9.217310407186341395835167909208

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