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Properties

Label 4-27936-1.1-c1e2-0-1
Degree $4$
Conductor $27936$
Sign $1$
Analytic cond. $1.78122$
Root an. cond. $1.15525$
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 + 8-s − 3·9-s + 8·11-s + 16-s − 3·18-s + 8·22-s − 4·23-s − 2·25-s + 32-s − 3·36-s − 4·37-s + 8·44-s − 4·46-s + 20·47-s − 10·49-s − 2·50-s + 4·59-s − 4·61-s + 64-s − 8·71-s − 3·72-s − 12·73-s − 4·74-s + 9·81-s − 8·83-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s + 2.41·11-s + 1/4·16-s − 0.707·18-s + 1.70·22-s − 0.834·23-s − 2/5·25-s + 0.176·32-s − 1/2·36-s − 0.657·37-s + 1.20·44-s − 0.589·46-s + 2.91·47-s − 1.42·49-s − 0.282·50-s + 0.520·59-s − 0.512·61-s + 1/8·64-s − 0.949·71-s − 0.353·72-s − 1.40·73-s − 0.464·74-s + 81-s − 0.878·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27936\)    =    \(2^{5} \cdot 3^{2} \cdot 97\)
Sign: $1$
Analytic conductor: \(1.78122\)
Root analytic conductor: \(1.15525\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 27936,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.927542009\)
\(L(\frac12)\) \(\approx\) \(1.927542009\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_2$ \( 1 + p T^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 10 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
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$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.19.a_abi
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.e_bu
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.31.a_as
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.e_o
41$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.41.a_g
43$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.43.a_g
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.47.au_hi
53$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.53.a_aba
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ae_eo
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.i_fm
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.m_gk
79$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.79.a_ak
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.89.a_aby
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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

−10.66359921521123238057919469814, −10.13261999899801358959842761084, −9.431953632445172003255668954476, −8.949684274449734132008103401204, −8.615257520851950705128539790068, −7.80873088584636847182917368360, −7.17776374540491363857963842977, −6.58938909036221452766181480006, −6.02276589655087689695686273515, −5.70432737408572891681371504948, −4.73574288193244504769188328185, −4.00239640281280857618712545439, −3.61778991240424903155027750169, −2.61430525590403854836049166267, −1.51079912547862319044955762489, 1.51079912547862319044955762489, 2.61430525590403854836049166267, 3.61778991240424903155027750169, 4.00239640281280857618712545439, 4.73574288193244504769188328185, 5.70432737408572891681371504948, 6.02276589655087689695686273515, 6.58938909036221452766181480006, 7.17776374540491363857963842977, 7.80873088584636847182917368360, 8.615257520851950705128539790068, 8.949684274449734132008103401204, 9.431953632445172003255668954476, 10.13261999899801358959842761084, 10.66359921521123238057919469814

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