↑

Properties

Label 4-93312-1.1-c1e2-0-35
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 9·11-s + 16-s − 3·17-s − 8·19-s − 9·22-s − 7·25-s + 32-s − 3·34-s − 8·38-s − 3·41-s + 43-s − 9·44-s + 8·49-s − 7·50-s + 9·59-s + 64-s − 2·67-s − 3·68-s − 8·73-s − 8·76-s − 3·82-s − 3·83-s + 86-s − 9·88-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2.71·11-s + 1/4·16-s − 0.727·17-s − 1.83·19-s − 1.91·22-s − 7/5·25-s + 0.176·32-s − 0.514·34-s − 1.29·38-s − 0.468·41-s + 0.152·43-s − 1.35·44-s + 8/7·49-s − 0.989·50-s + 1.17·59-s + 1/8·64-s − 0.244·67-s − 0.363·68-s − 0.936·73-s − 0.917·76-s − 0.331·82-s − 0.329·83-s + 0.107·86-s − 0.959·88-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: \(1\)
Selberg data: \((4,\ 93312,\ (\ :1/2, 1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( 1 - T \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
11$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.j_bo
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.i_bt
23$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \) 2.23.a_abd
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.29.a_aba
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.37.a_ao
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.d_de
43$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ab_co
47$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.47.a_ar
53$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.53.a_dh
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.aj_eo
61$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.61.a_cy
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.c_ew
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.71.a_aby
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.i_s
79$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.79.a_abg
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.d_ei
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ap_gw
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.97.ae_gg
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

−9.466735420720199237667881028373, −8.632486549622069058973429947631, −8.400531122375489278561384205409, −7.78052733578435752873905658020, −7.44553777356754761019567442693, −6.79406350907268642327012460387, −6.16648379226767935948841723392, −5.65538046868817425384744042033, −5.22014038378074018119269838350, −4.56159315065695703185774077470, −4.10263164029946493874977314399, −3.24226054134041403316112910828, −2.35224469819524360178368958502, −2.17811480859044972128444750018, 0, 2.17811480859044972128444750018, 2.35224469819524360178368958502, 3.24226054134041403316112910828, 4.10263164029946493874977314399, 4.56159315065695703185774077470, 5.22014038378074018119269838350, 5.65538046868817425384744042033, 6.16648379226767935948841723392, 6.79406350907268642327012460387, 7.44553777356754761019567442693, 7.78052733578435752873905658020, 8.400531122375489278561384205409, 8.632486549622069058973429947631, 9.466735420720199237667881028373

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