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Properties

Label 4-197568-1.1-c1e2-0-4
Degree $4$
Conductor $197568$
Sign $1$
Analytic cond. $12.5971$
Root an. cond. $1.88394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 7-s − 3·8-s − 9-s − 14-s − 16-s − 18-s − 2·25-s + 28-s − 8·29-s + 5·32-s + 36-s + 8·37-s + 8·43-s + 49-s − 2·50-s + 16·53-s + 3·56-s − 8·58-s + 63-s + 7·64-s + 8·71-s + 3·72-s + 8·74-s + 16·79-s + 81-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 − 2/5·25-s + 0.188·28-s − 1.48·29-s + 0.883·32-s + 1/6·36-s + 1.31·37-s + 1.21·43-s + 1/7·49-s − 0.282·50-s + 2.19·53-s + 0.400·56-s − 1.05·58-s + 0.125·63-s + 7/8·64-s + 0.949·71-s + 0.353·72-s + 0.929·74-s + 1.80·79-s + 1/9·81-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

\(L(1)\) \(\approx\) \(1.499275166\)
\(L(\frac12)\) \(\approx\) \(1.499275166\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.i_bm
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.ai_di
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.41.a_aby
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.47.a_abi
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.aq_gk
59$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.59.a_abm
61$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.61.a_bi
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ai_fm
73$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.73.a_o
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) 2.79.aq_gc
83$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.83.a_cw
89$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.89.a_be
97$C_2^2$ \( 1 + 174 T^{2} + p^{2} T^{4} \) 2.97.a_gs
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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

−9.238879676310683936239309686272, −8.715491847344747864034263141343, −8.058444420985550901489535521991, −7.79961963051811672837280599905, −7.06658899531469629692365034640, −6.61629162482949800133275291791, −5.90975680274322949063473683183, −5.66985875865153575451317779965, −5.20295829665866836989138489172, −4.43167623125588569125019680873, −3.98907773898537122548418353024, −3.52291225486626047826688607349, −2.77502878580488056295383711588, −2.13693696493325132336260039083, −0.69540681595204814212171147456, 0.69540681595204814212171147456, 2.13693696493325132336260039083, 2.77502878580488056295383711588, 3.52291225486626047826688607349, 3.98907773898537122548418353024, 4.43167623125588569125019680873, 5.20295829665866836989138489172, 5.66985875865153575451317779965, 5.90975680274322949063473683183, 6.61629162482949800133275291791, 7.06658899531469629692365034640, 7.79961963051811672837280599905, 8.058444420985550901489535521991, 8.715491847344747864034263141343, 9.238879676310683936239309686272

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