↑

Properties

Label 4-93312-1.1-c1e2-0-6
Degree $4$
Conductor $93312$
Sign $1$
Analytic cond. $5.94965$
Root an. cond. $1.56179$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 3·11-s + 12-s + 16-s + 9·17-s − 18-s + 19-s − 3·22-s − 24-s + 2·25-s + 27-s − 32-s + 3·33-s − 9·34-s + 36-s − 38-s + 6·41-s − 11·43-s + 3·44-s + 48-s − 13·49-s − 2·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 1/4·16-s + 2.18·17-s − 0.235·18-s + 0.229·19-s − 0.639·22-s − 0.204·24-s + 2/5·25-s + 0.192·27-s − 0.176·32-s + 0.522·33-s − 1.54·34-s + 1/6·36-s − 0.162·38-s + 0.937·41-s − 1.67·43-s + 0.452·44-s + 0.144·48-s − 1.85·49-s − 0.282·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(93312\)    =    \(2^{7} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(5.94965\)
Root analytic conductor: \(1.56179\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 93312,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.537778986\)
\(L(\frac12)\) \(\approx\) \(1.537778986\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.ad_e
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.13.a_ai
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.17.aj_ca
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.ab_s
23$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.23.a_z
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.29.a_bu
31$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.31.a_abj
37$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.37.a_q
41$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.ag_cd
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.l_ds
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.47.a_bu
53$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.53.a_w
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.59.d_eo
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.61.a_aby
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.67.l_ee
71$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.71.a_adi
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.ab_bk
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.79.a_fa
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.83.ad_fs
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.747707191175209441218296822071, −9.113408463243647775134903010279, −8.695348607584860387911326116543, −8.123028121475115051227077007196, −7.70776955757295169857288579509, −7.34800435577449555843449973696, −6.61839689115813819100372203397, −6.23252991734223231797622937812, −5.52545831247434625509168219034, −4.92310656236982106258656946630, −4.12786989261980462344842823209, −3.27652432740011333824788859201, −3.09975095477689994772071629147, −1.85278941636313118481788942622, −1.13139588167605795155122494146, 1.13139588167605795155122494146, 1.85278941636313118481788942622, 3.09975095477689994772071629147, 3.27652432740011333824788859201, 4.12786989261980462344842823209, 4.92310656236982106258656946630, 5.52545831247434625509168219034, 6.23252991734223231797622937812, 6.61839689115813819100372203397, 7.34800435577449555843449973696, 7.70776955757295169857288579509, 8.123028121475115051227077007196, 8.695348607584860387911326116543, 9.113408463243647775134903010279, 9.747707191175209441218296822071

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