Properties

Label 4-33e4-1.1-c1e2-0-19
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  + 3·4-s + 4·7-s + 10·13-s + 5·16-s − 3·25-s + 12·28-s + 8·31-s − 6·37-s + 12·43-s − 2·49-s + 30·52-s − 12·61-s + 3·64-s − 20·67-s − 20·73-s − 16·79-s + 40·91-s + 10·97-s − 9·100-s + 28·103-s + 10·109-s + 20·112-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 3/2·4-s + 1.51·7-s + 2.77·13-s + 5/4·16-s − 3/5·25-s + 2.26·28-s + 1.43·31-s − 0.986·37-s + 1.82·43-s − 2/7·49-s + 4.16·52-s − 1.53·61-s + 3/8·64-s − 2.44·67-s − 2.34·73-s − 1.80·79-s + 4.19·91-s + 1.01·97-s − 0.899·100-s + 2.75·103-s + 0.957·109-s + 1.88·112-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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

\(L(1)\) \(\approx\) \(4.909407779\)
\(L(\frac12)\) \(\approx\) \(4.909407779\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 27 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 75 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
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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

−10.42050543841446949014843697780, −9.823301014418404707615408714952, −8.888118693639745723205299374602, −8.779446533560007743599347167638, −8.569574567281581248330849944299, −7.77594574784300739537262892995, −7.66517707877848537405587208908, −7.35922998547033483473724572447, −6.55220249854625099979645502718, −6.16295820843450353391913019912, −6.08939320500831878326768482673, −5.56139704051060845600743152726, −4.87924307440615116349372274396, −4.29707997991328737318830135800, −3.99480843407406712831031412201, −3.03439835762242077521563939459, −3.01381741469607397174690710148, −1.90608658468002312089244101013, −1.59089441543721140954265628235, −1.12509458117987250833598923208, 1.12509458117987250833598923208, 1.59089441543721140954265628235, 1.90608658468002312089244101013, 3.01381741469607397174690710148, 3.03439835762242077521563939459, 3.99480843407406712831031412201, 4.29707997991328737318830135800, 4.87924307440615116349372274396, 5.56139704051060845600743152726, 6.08939320500831878326768482673, 6.16295820843450353391913019912, 6.55220249854625099979645502718, 7.35922998547033483473724572447, 7.66517707877848537405587208908, 7.77594574784300739537262892995, 8.569574567281581248330849944299, 8.779446533560007743599347167638, 8.888118693639745723205299374602, 9.823301014418404707615408714952, 10.42050543841446949014843697780

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