Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 8·11-s − 6·25-s + 27-s + 8·33-s + 10·49-s + 8·59-s + 20·73-s − 6·75-s + 81-s − 24·83-s + 20·97-s + 8·99-s − 8·107-s + 26·121-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 2.41·11-s − 6/5·25-s + 0.192·27-s + 1.39·33-s + 10/7·49-s + 1.04·59-s + 2.34·73-s − 0.692·75-s + 1/9·81-s − 2.63·83-s + 2.03·97-s + 0.804·99-s − 0.773·107-s + 2.36·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(221184\)    =    \(2^{13} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(14.1028\)
Root analytic conductor: \(1.93788\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 221184,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.516665776\)
\(L(\frac12)\) \(\approx\) \(2.516665776\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.a_acw
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.a_as
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−9.084060710514600119498421168942, −8.682903708494999446134722434078, −8.115297202153064787735995185656, −7.65984528316703258377451127493, −6.99427189060156094669299763753, −6.76837689568436453091584640890, −6.16789832421500606250011960556, −5.72769471315477268204103802164, −5.00579154045731857259636739405, −4.22625152898219813111253915900, −3.89884721747839950717610886390, −3.51715563036898802866811855017, −2.56140637552839117703673801031, −1.83678997807251975324842072510, −1.06083089704810257608250345977, 1.06083089704810257608250345977, 1.83678997807251975324842072510, 2.56140637552839117703673801031, 3.51715563036898802866811855017, 3.89884721747839950717610886390, 4.22625152898219813111253915900, 5.00579154045731857259636739405, 5.72769471315477268204103802164, 6.16789832421500606250011960556, 6.76837689568436453091584640890, 6.99427189060156094669299763753, 7.65984528316703258377451127493, 8.115297202153064787735995185656, 8.682903708494999446134722434078, 9.084060710514600119498421168942

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