Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 7-s − 3·8-s − 9-s + 14-s − 16-s − 18-s + 6·25-s − 28-s + 5·32-s + 36-s + 49-s + 6·50-s − 3·56-s − 63-s + 7·64-s + 3·72-s + 32·79-s + 81-s + 98-s − 6·100-s − 112-s + 28·113-s − 6·121-s − 126-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 1/3·9-s + 0.267·14-s − 1/4·16-s − 0.235·18-s + 6/5·25-s − 0.188·28-s + 0.883·32-s + 1/6·36-s + 1/7·49-s + 0.848·50-s − 0.400·56-s − 0.125·63-s + 7/8·64-s + 0.353·72-s + 3.60·79-s + 1/9·81-s + 0.101·98-s − 3/5·100-s − 0.0944·112-s + 2.63·113-s − 0.545·121-s − 0.0890·126-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(197568\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(12.5971\)
Root analytic conductor: \(1.88394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 197568,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.843198006\)
\(L(\frac12)\) \(\approx\) \(1.843198006\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2$C_2$ \( 1 - T + p T^{2} \)
3$C_2$ \( 1 + T^{2} \)
7$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.59.a_ba
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.61.a_aeo
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.a_as
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−9.088296158561309606146655867375, −8.702055503069812456806332641439, −8.116084574615389597608747687913, −7.82142617407106240780576661084, −7.06694747197912778558188301570, −6.57637561281110173737978138450, −6.09038838465970260046484749754, −5.51934697089649114897831128111, −5.00826014686393218390262936776, −4.68409389886683540315675965665, −4.00085365664919974852022577273, −3.41745442814838319552427443793, −2.85374809364316161963552276272, −2.04048845845385650205785524852, −0.793834388843266153539947764495, 0.793834388843266153539947764495, 2.04048845845385650205785524852, 2.85374809364316161963552276272, 3.41745442814838319552427443793, 4.00085365664919974852022577273, 4.68409389886683540315675965665, 5.00826014686393218390262936776, 5.51934697089649114897831128111, 6.09038838465970260046484749754, 6.57637561281110173737978138450, 7.06694747197912778558188301570, 7.82142617407106240780576661084, 8.116084574615389597608747687913, 8.702055503069812456806332641439, 9.088296158561309606146655867375

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