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Properties

Label 4-6e8-1.1-c1e2-0-33
Degree $4$
Conductor $1679616$
Sign $1$
Analytic cond. $107.093$
Root an. cond. $3.21692$
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  − 3·5-s + 2·7-s − 6·11-s − 5·13-s − 6·17-s − 4·19-s + 6·23-s + 5·25-s − 3·29-s − 4·31-s − 6·35-s + 10·37-s + 6·41-s − 10·43-s + 7·49-s − 12·53-s + 18·55-s − 12·59-s − 5·61-s + 15·65-s + 2·67-s − 12·71-s − 2·73-s − 12·77-s − 10·79-s + 18·85-s − 6·89-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.755·7-s − 1.80·11-s − 1.38·13-s − 1.45·17-s − 0.917·19-s + 1.25·23-s + 25-s − 0.557·29-s − 0.718·31-s − 1.01·35-s + 1.64·37-s + 0.937·41-s − 1.52·43-s + 49-s − 1.64·53-s + 2.42·55-s − 1.56·59-s − 0.640·61-s + 1.86·65-s + 0.244·67-s − 1.42·71-s − 0.234·73-s − 1.36·77-s − 1.12·79-s + 1.95·85-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(107.093\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_e
7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_ad
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_z
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.f_m
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_au
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.e_ap
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.37.ak_dv
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_af
43$C_2^2$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_cf
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_abk
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_acl
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.73.c_fr
79$C_2^2$ \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_v
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.89.g_hf
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_d
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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

−9.259650295228427430645924809421, −9.113352002665881162914952293930, −8.577333685401399343234855340031, −7.970110469822135180309382130496, −7.84021219639205210408512312379, −7.68876969644185879207134511653, −6.95530601917106787739119657215, −6.87275115150276690653262973933, −6.14522234673339724413771065041, −5.50199424166099694755149716467, −5.10290378074997383625600015200, −4.72026964119258487434264193119, −4.30982852181024666686001526412, −4.07300718772695014351583245664, −2.97092318778329899567688534320, −2.84601146801517810354973447351, −2.25622874780795178387995546928, −1.47282272623686094791119897549, 0, 0, 1.47282272623686094791119897549, 2.25622874780795178387995546928, 2.84601146801517810354973447351, 2.97092318778329899567688534320, 4.07300718772695014351583245664, 4.30982852181024666686001526412, 4.72026964119258487434264193119, 5.10290378074997383625600015200, 5.50199424166099694755149716467, 6.14522234673339724413771065041, 6.87275115150276690653262973933, 6.95530601917106787739119657215, 7.68876969644185879207134511653, 7.84021219639205210408512312379, 7.970110469822135180309382130496, 8.577333685401399343234855340031, 9.113352002665881162914952293930, 9.259650295228427430645924809421

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