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Properties

Label 4-42e4-1.1-c1e2-0-19
Degree $4$
Conductor $3111696$
Sign $1$
Analytic cond. $198.404$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 6·5-s + 6·9-s + 6·11-s + 13-s + 18·15-s − 6·17-s + 4·19-s − 6·23-s + 17·25-s + 9·27-s − 3·29-s − 5·31-s + 18·33-s − 2·37-s + 3·39-s − 3·41-s + 43-s + 36·45-s + 9·47-s − 18·51-s + 6·53-s + 36·55-s + 12·57-s + 3·59-s + 13·61-s + 6·65-s + ⋯
L(s)  = 1  + 1.73·3-s + 2.68·5-s + 2·9-s + 1.80·11-s + 0.277·13-s + 4.64·15-s − 1.45·17-s + 0.917·19-s − 1.25·23-s + 17/5·25-s + 1.73·27-s − 0.557·29-s − 0.898·31-s + 3.13·33-s − 0.328·37-s + 0.480·39-s − 0.468·41-s + 0.152·43-s + 5.36·45-s + 1.31·47-s − 2.52·51-s + 0.824·53-s + 4.85·55-s + 1.58·57-s + 0.390·59-s + 1.66·61-s + 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(198.404\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3111696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.957875352\)
\(L(\frac12)\) \(\approx\) \(9.957875352\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - p T + p T^{2} \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_am
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_t
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_ad
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.23.g_cd
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_au
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_ag
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_bi
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.59.ad_aby
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.an_ee
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.67.ah_as
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.ak_bb
79$C_2^2$ \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.79.l_bq
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_acb
97$C_2^2$ \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.97.l_y
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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

−9.521639218601219668900771416064, −9.119132060948563196077013066792, −8.760153169971804146104741828183, −8.690412663272820784310503814405, −8.093883675189657535525462497639, −7.44591564765791901932394967454, −6.99016945257227287322964106892, −6.86019793862356855528174165572, −6.16754609675195071409399296125, −6.07132490515510078279087567771, −5.46822607169730739532048724547, −5.17679836486865466942365181029, −4.21197945409123824725531504680, −4.10053570936901699772942819289, −3.58315320934921240019072276239, −2.94680238624405992046525258292, −2.24975969097695172950960929349, −2.13830282938126186905957234257, −1.64474442766836725824304879449, −1.17537700419931611653622631589, 1.17537700419931611653622631589, 1.64474442766836725824304879449, 2.13830282938126186905957234257, 2.24975969097695172950960929349, 2.94680238624405992046525258292, 3.58315320934921240019072276239, 4.10053570936901699772942819289, 4.21197945409123824725531504680, 5.17679836486865466942365181029, 5.46822607169730739532048724547, 6.07132490515510078279087567771, 6.16754609675195071409399296125, 6.86019793862356855528174165572, 6.99016945257227287322964106892, 7.44591564765791901932394967454, 8.093883675189657535525462497639, 8.690412663272820784310503814405, 8.760153169971804146104741828183, 9.119132060948563196077013066792, 9.521639218601219668900771416064

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