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Properties

Label 4-1769472-1.1-c1e2-0-60
Degree $4$
Conductor $1769472$
Sign $-1$
Analytic cond. $112.823$
Root an. cond. $3.25911$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 6·17-s − 8·19-s + 6·25-s + 27-s + 6·41-s − 12·43-s − 2·49-s + 6·51-s − 8·57-s − 12·59-s − 8·67-s − 4·73-s + 6·75-s + 81-s − 24·83-s − 30·89-s + 16·97-s − 12·107-s + 18·113-s − 18·121-s + 6·123-s + 127-s − 12·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.45·17-s − 1.83·19-s + 6/5·25-s + 0.192·27-s + 0.937·41-s − 1.82·43-s − 2/7·49-s + 0.840·51-s − 1.05·57-s − 1.56·59-s − 0.977·67-s − 0.468·73-s + 0.692·75-s + 1/9·81-s − 2.63·83-s − 3.17·89-s + 1.62·97-s − 1.16·107-s + 1.69·113-s − 1.63·121-s + 0.541·123-s + 0.0887·127-s − 1.05·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1769472\)    =    \(2^{16} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(112.823\)
Root analytic conductor: \(3.25911\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1769472,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ag_bq
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.29.a_ao
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ag_de
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.m_di
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.m_fu
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.71.a_bi
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.e_g
79$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.79.a_abu
83$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.83.y_ko
89$C_2$$\times$$C_2$ \( ( 1 + 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.be_pm
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.aq_io
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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

−7.62896499680929458799920735992, −7.17706921176233681175606330955, −6.87287392161813965905405658978, −6.29160274849111367191460609561, −5.97233479206907421582725276803, −5.45096928640795251936350835258, −4.89538338080886372154356995257, −4.43004081427224699344935975670, −4.08746208433189831832110712424, −3.38576421785665613978219138346, −2.98510059872765653171712643753, −2.54202091418068344891901363477, −1.69640001968282789835812096363, −1.25734401641383984354489752394, 0, 1.25734401641383984354489752394, 1.69640001968282789835812096363, 2.54202091418068344891901363477, 2.98510059872765653171712643753, 3.38576421785665613978219138346, 4.08746208433189831832110712424, 4.43004081427224699344935975670, 4.89538338080886372154356995257, 5.45096928640795251936350835258, 5.97233479206907421582725276803, 6.29160274849111367191460609561, 6.87287392161813965905405658978, 7.17706921176233681175606330955, 7.62896499680929458799920735992

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