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Properties

Label 4-3833280-1.1-c1e2-0-2
Degree $4$
Conductor $3833280$
Sign $1$
Analytic cond. $244.413$
Root an. cond. $3.95395$
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  + 3·5-s − 3·9-s − 11-s − 4·23-s + 2·25-s − 4·31-s + 4·37-s − 9·45-s + 8·47-s − 2·49-s + 12·53-s − 3·55-s − 24·59-s − 12·67-s − 16·71-s + 9·81-s + 12·97-s + 3·99-s + 16·103-s + 28·113-s − 12·115-s + 121-s − 10·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.34·5-s − 9-s − 0.301·11-s − 0.834·23-s + 2/5·25-s − 0.718·31-s + 0.657·37-s − 1.34·45-s + 1.16·47-s − 2/7·49-s + 1.64·53-s − 0.404·55-s − 3.12·59-s − 1.46·67-s − 1.89·71-s + 81-s + 1.21·97-s + 0.301·99-s + 1.57·103-s + 2.63·113-s − 1.11·115-s + 1/11·121-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3833280 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3833280 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3833280\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(244.413\)
Root analytic conductor: \(3.95395\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3833280,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.946321759\)
\(L(\frac12)\) \(\approx\) \(1.946321759\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
11$C_1$ \( 1 + T \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.e_o
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.e_ck
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.41.a_o
43$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.43.a_s
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ai_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.m_gk
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.q_fm
73$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.73.a_o
79$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.79.a_c
83$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.83.a_cs
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.am_di
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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

−7.44241288704876535407746120731, −7.15394411786807019027066626539, −6.29095426404520384220001738867, −6.14757502439302196772219335011, −5.83649563659123283149367530023, −5.59127212919782325711928790832, −4.96496255437477024170852905362, −4.58441829787946304776712431178, −4.06898835471629482195152746644, −3.39980035364627565129724837053, −2.96913996159594065067180192225, −2.47073978269101576753217214350, −1.97248740060207491400412322511, −1.51099445052403697852558205977, −0.48470162605937222560287976105, 0.48470162605937222560287976105, 1.51099445052403697852558205977, 1.97248740060207491400412322511, 2.47073978269101576753217214350, 2.96913996159594065067180192225, 3.39980035364627565129724837053, 4.06898835471629482195152746644, 4.58441829787946304776712431178, 4.96496255437477024170852905362, 5.59127212919782325711928790832, 5.83649563659123283149367530023, 6.14757502439302196772219335011, 6.29095426404520384220001738867, 7.15394411786807019027066626539, 7.44241288704876535407746120731

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