Normalization:  

Dirichlet series

L(s)  = 1  − 6·7-s − 4·16-s + 19-s + 9·25-s − 2·43-s + 13·49-s − 14·61-s − 22·73-s + 24·112-s + 13·121-s + ⋯
L(s)  = 1  − 2.26·7-s − 16-s + 0.229·19-s + 9/5·25-s − 0.304·43-s + 13/7·49-s − 1.79·61-s − 2.57·73-s + 2.26·112-s + 1.18·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(555579\)    =    \(3^{4} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(35.4241\)
Root analytic conductor: \(2.43963\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 555579,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8312980978\)
\(L(\frac12)\) \(\approx\) \(0.8312980978\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
5$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.5.a_aj
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.11.a_an
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.17.a_az
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.a_cg
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.47.a_adh
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.61.o_gp
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−8.654665863086876796946262541530, −8.021221283894327547248318996257, −7.33265999401317932020908373439, −7.00986178656700423544432429669, −6.68550161853144269147197062723, −6.19576308552835465873859700522, −5.91089219284319100501059372949, −5.19338516815505833883713817521, −4.61063282324966903504187586929, −4.17923218777483676937115691602, −3.35595717755380667916779766783, −3.05631573153132411893979180784, −2.65803781473737374352150912745, −1.63526370260409934846124451980, −0.46177192895213506821070958612, 0.46177192895213506821070958612, 1.63526370260409934846124451980, 2.65803781473737374352150912745, 3.05631573153132411893979180784, 3.35595717755380667916779766783, 4.17923218777483676937115691602, 4.61063282324966903504187586929, 5.19338516815505833883713817521, 5.91089219284319100501059372949, 6.19576308552835465873859700522, 6.68550161853144269147197062723, 7.00986178656700423544432429669, 7.33265999401317932020908373439, 8.021221283894327547248318996257, 8.654665863086876796946262541530

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