Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·9-s − 9·25-s − 5·27-s + 18·31-s + 10·37-s − 2·49-s + 22·67-s − 9·75-s + 81-s + 18·93-s + 2·97-s + 8·103-s + 10·111-s − 11·121-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s − 9/5·25-s − 0.962·27-s + 3.23·31-s + 1.64·37-s − 2/7·49-s + 2.68·67-s − 1.03·75-s + 1/9·81-s + 1.86·93-s + 0.203·97-s + 0.788·103-s + 0.949·111-s − 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(278784\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(17.7755\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 278784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.909133079\)
\(L(\frac12)\) \(\approx\) \(1.909133079\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.31.as_fn
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.37.ak_dv
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.a_ack
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.59.a_cr
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.67.aw_jv
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.a_cj
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.79.a_adq
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.89.a_gv
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.97.ac_hn
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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

−8.807420656810522908555641840351, −8.296216156067471490779788750581, −7.917551266650708038064411233522, −7.83935802506486901627808754804, −6.97891364292088590488920489309, −6.38938507877913915880331745407, −6.12208142017703739786852081048, −5.54098847555261772857433961580, −4.93410667015016317105118211163, −4.29729803577494927843964511661, −3.86640013558637673014838155229, −3.09343609976673865899886574907, −2.59981942012243216712947457866, −2.00464930491110072965238708177, −0.796122916422569527316567043458, 0.796122916422569527316567043458, 2.00464930491110072965238708177, 2.59981942012243216712947457866, 3.09343609976673865899886574907, 3.86640013558637673014838155229, 4.29729803577494927843964511661, 4.93410667015016317105118211163, 5.54098847555261772857433961580, 6.12208142017703739786852081048, 6.38938507877913915880331745407, 6.97891364292088590488920489309, 7.83935802506486901627808754804, 7.917551266650708038064411233522, 8.296216156067471490779788750581, 8.807420656810522908555641840351

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