Properties

Label 4-1110e2-1.1-c1e2-0-11
Degree $4$
Conductor $1232100$
Sign $1$
Analytic cond. $78.5597$
Root an. cond. $2.97714$
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  − 2·2-s + 3·4-s + 4·5-s − 4·8-s − 9-s − 8·10-s − 6·11-s − 8·13-s + 5·16-s + 14·17-s + 2·18-s + 12·20-s + 12·22-s + 12·23-s + 11·25-s + 16·26-s − 6·32-s − 28·34-s − 3·36-s + 12·37-s − 16·40-s + 14·41-s + 2·43-s − 18·44-s − 4·45-s − 24·46-s + 5·49-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.78·5-s − 1.41·8-s − 1/3·9-s − 2.52·10-s − 1.80·11-s − 2.21·13-s + 5/4·16-s + 3.39·17-s + 0.471·18-s + 2.68·20-s + 2.55·22-s + 2.50·23-s + 11/5·25-s + 3.13·26-s − 1.06·32-s − 4.80·34-s − 1/2·36-s + 1.97·37-s − 2.52·40-s + 2.18·41-s + 0.304·43-s − 2.71·44-s − 0.596·45-s − 3.53·46-s + 5/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1232100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(78.5597\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1110} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1232100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.614357300\)
\(L(\frac12)\) \(\approx\) \(1.614357300\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
37$C_2$ \( 1 - 12 T + p T^{2} \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−9.944785342426537184050889714096, −9.523444578773864700398708753408, −9.464422765222231864240239895120, −9.142063290605765694918676228180, −8.338064904277218959335980431427, −7.944800691313305294392262098999, −7.56616110415167812477128210763, −7.43135631203755985745287464329, −6.92438416930772307260789120583, −6.27035422248702251553430672134, −5.62898898145511302419375314104, −5.61658549998341156981163708296, −5.01645746251138040909617075311, −4.88171201275687955830839733343, −3.51741779286829605121937274790, −2.70870410307633660129651567510, −2.67749981244277732549187936917, −2.36715974832215047004468296495, −1.17622194661047059541889486780, −0.824332873162489645089156544143, 0.824332873162489645089156544143, 1.17622194661047059541889486780, 2.36715974832215047004468296495, 2.67749981244277732549187936917, 2.70870410307633660129651567510, 3.51741779286829605121937274790, 4.88171201275687955830839733343, 5.01645746251138040909617075311, 5.61658549998341156981163708296, 5.62898898145511302419375314104, 6.27035422248702251553430672134, 6.92438416930772307260789120583, 7.43135631203755985745287464329, 7.56616110415167812477128210763, 7.944800691313305294392262098999, 8.338064904277218959335980431427, 9.142063290605765694918676228180, 9.464422765222231864240239895120, 9.523444578773864700398708753408, 9.944785342426537184050889714096

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