Properties

Label 4-1932e2-1.1-c1e2-0-6
Degree $4$
Conductor $3732624$
Sign $1$
Analytic cond. $237.995$
Root an. cond. $3.92773$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s − 2·7-s + 3·9-s − 4·11-s + 3·13-s − 2·15-s − 2·19-s + 4·21-s + 2·23-s − 8·25-s − 4·27-s + 2·29-s − 8·31-s + 8·33-s − 2·35-s − 2·37-s − 6·39-s + 6·41-s − 5·43-s + 3·45-s − 6·47-s + 3·49-s − 3·53-s − 4·55-s + 4·57-s − 9·59-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s − 0.755·7-s + 9-s − 1.20·11-s + 0.832·13-s − 0.516·15-s − 0.458·19-s + 0.872·21-s + 0.417·23-s − 8/5·25-s − 0.769·27-s + 0.371·29-s − 1.43·31-s + 1.39·33-s − 0.338·35-s − 0.328·37-s − 0.960·39-s + 0.937·41-s − 0.762·43-s + 0.447·45-s − 0.875·47-s + 3/7·49-s − 0.412·53-s − 0.539·55-s + 0.529·57-s − 1.17·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3732624\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(237.995\)
Root analytic conductor: \(3.92773\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1932} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3732624,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 3 T + 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 73 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 5 T - 9 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 9 T + 137 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 5 T + 67 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 9 T + 153 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 15 T + 187 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 189 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T + 87 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - T + 167 T^{2} - p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 193 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−8.932526094701324319897592091112, −8.835997115331228201459127588499, −7.976278093522204236014832747718, −7.940875296716682488891264803776, −7.22458274767723424721983952468, −7.12560688888952564741987005992, −6.45173771275297721777090503583, −6.07568979709678840556042024904, −5.84197714289754961492890811542, −5.61287756590038162270844640024, −4.88057864523329737380107176582, −4.76293495531927825566473534545, −3.85544166182386658974120333312, −3.84059241710182488095682273370, −2.87585854646542638452068144525, −2.69648303044189013539176822333, −1.62857204923978028627735473441, −1.47619492191824786851912018750, 0, 0, 1.47619492191824786851912018750, 1.62857204923978028627735473441, 2.69648303044189013539176822333, 2.87585854646542638452068144525, 3.84059241710182488095682273370, 3.85544166182386658974120333312, 4.76293495531927825566473534545, 4.88057864523329737380107176582, 5.61287756590038162270844640024, 5.84197714289754961492890811542, 6.07568979709678840556042024904, 6.45173771275297721777090503583, 7.12560688888952564741987005992, 7.22458274767723424721983952468, 7.940875296716682488891264803776, 7.976278093522204236014832747718, 8.835997115331228201459127588499, 8.932526094701324319897592091112

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