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Properties

Label 4-444e2-1.1-c1e2-0-3
Degree $4$
Conductor $197136$
Sign $1$
Analytic cond. $12.5695$
Root an. cond. $1.88291$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 6·5-s − 7-s + 12·11-s + 3·13-s + 6·15-s + 12·17-s + 6·19-s + 21-s + 19·25-s + 27-s − 12·33-s + 6·35-s − 10·37-s − 3·39-s − 12·41-s + 24·47-s + 7·49-s − 12·51-s − 6·53-s − 72·55-s − 6·57-s − 6·59-s + 12·61-s − 18·65-s + 5·67-s + 14·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 2.68·5-s − 0.377·7-s + 3.61·11-s + 0.832·13-s + 1.54·15-s + 2.91·17-s + 1.37·19-s + 0.218·21-s + 19/5·25-s + 0.192·27-s − 2.08·33-s + 1.01·35-s − 1.64·37-s − 0.480·39-s − 1.87·41-s + 3.50·47-s + 49-s − 1.68·51-s − 0.824·53-s − 9.70·55-s − 0.794·57-s − 0.781·59-s + 1.53·61-s − 2.23·65-s + 0.610·67-s + 1.63·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(197136\)    =    \(2^{4} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(12.5695\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 197136,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.206827895\)
\(L(\frac12)\) \(\approx\) \(1.206827895\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
37$C_2$ \( 1 + 10 T + p T^{2} \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.5.g_r
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.b_ag
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.11.am_cg
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.ad_q
17$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.17.am_cn
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ag_bf
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.a_ach
41$C_2^2$ \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_dz
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.a_cj
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.47.ay_je
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_ct
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.am_ef
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.af_abq
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.71.a_act
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.73.ao_hn
79$C_2^2$ \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.79.ad_de
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2^2$ \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_hp
97$C_2^2$ \( 1 - 191 T^{2} + p^{2} T^{4} \) 2.97.a_ahj
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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

−11.44073481090929133702128067963, −11.24320851060650694253853204392, −10.51242375931511721908905377649, −10.09715224295158755465725677084, −9.282961808713625333709773574177, −9.208022976846388863639716569141, −8.516335226941541264038851481628, −8.190788651425422387238510747981, −7.49083261947874685092494477050, −7.28926154674775956289824025102, −6.75098917075190470870617041203, −6.36881781786042333648821517624, −5.58050717714227387758351222668, −5.19058648214797807514402718648, −4.13476642381958182566373018637, −3.89203993851820244067982668909, −3.43762632457338451517636771204, −3.39698377254026474132856635670, −1.19131287461873159632462798164, −0.976423742954294728741394703995, 0.976423742954294728741394703995, 1.19131287461873159632462798164, 3.39698377254026474132856635670, 3.43762632457338451517636771204, 3.89203993851820244067982668909, 4.13476642381958182566373018637, 5.19058648214797807514402718648, 5.58050717714227387758351222668, 6.36881781786042333648821517624, 6.75098917075190470870617041203, 7.28926154674775956289824025102, 7.49083261947874685092494477050, 8.190788651425422387238510747981, 8.516335226941541264038851481628, 9.208022976846388863639716569141, 9.282961808713625333709773574177, 10.09715224295158755465725677084, 10.51242375931511721908905377649, 11.24320851060650694253853204392, 11.44073481090929133702128067963

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