Properties

Label 4-33e4-1.1-c1e2-0-7
Degree $4$
Conductor $1185921$
Sign $1$
Analytic cond. $75.6153$
Root an. cond. $2.94884$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 2·4-s − 5-s − 3·6-s + 4·7-s + 5·8-s + 6·9-s − 10-s − 6·12-s − 2·13-s + 4·14-s + 3·15-s + 5·16-s − 8·17-s + 6·18-s − 12·19-s − 2·20-s − 12·21-s + 4·23-s − 15·24-s + 5·25-s − 2·26-s − 9·27-s + 8·28-s + 6·29-s + 3·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 4-s − 0.447·5-s − 1.22·6-s + 1.51·7-s + 1.76·8-s + 2·9-s − 0.316·10-s − 1.73·12-s − 0.554·13-s + 1.06·14-s + 0.774·15-s + 5/4·16-s − 1.94·17-s + 1.41·18-s − 2.75·19-s − 0.447·20-s − 2.61·21-s + 0.834·23-s − 3.06·24-s + 25-s − 0.392·26-s − 1.73·27-s + 1.51·28-s + 1.11·29-s + 0.547·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(75.6153\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1185921,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.109197475\)
\(L(\frac12)\) \(\approx\) \(2.109197475\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + p T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
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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

−10.43948540498682382987905312962, −10.10415904465397542913543204554, −9.093319561592022991417688087591, −8.845751675703985759968870215611, −8.300417556931604842921731301395, −7.86301902968550665147629319634, −7.25668812088417939252798353338, −7.17438753673190165301444768888, −6.59510235598678484096179378227, −6.22012149996149691185417770290, −6.00767223225028104562166575000, −4.99273174447983405552720401135, −4.84872332465053519567303589009, −4.57970630690062027615572962378, −4.36019256832316784900076968449, −3.71017380830295321975015457814, −2.50204123203757374863124919182, −2.12704371623891776278186790072, −1.59487807486734172475132795992, −0.64563431816277849755901893368, 0.64563431816277849755901893368, 1.59487807486734172475132795992, 2.12704371623891776278186790072, 2.50204123203757374863124919182, 3.71017380830295321975015457814, 4.36019256832316784900076968449, 4.57970630690062027615572962378, 4.84872332465053519567303589009, 4.99273174447983405552720401135, 6.00767223225028104562166575000, 6.22012149996149691185417770290, 6.59510235598678484096179378227, 7.17438753673190165301444768888, 7.25668812088417939252798353338, 7.86301902968550665147629319634, 8.300417556931604842921731301395, 8.845751675703985759968870215611, 9.093319561592022991417688087591, 10.10415904465397542913543204554, 10.43948540498682382987905312962

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