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Properties

Label 4-6e8-1.1-c1e2-0-19
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 $0$

Origins

Origins of factors

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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: \(0\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.648399641\)
\(L(\frac12)\) \(\approx\) \(3.648399641\)
\(L(\frac{3}{2})\) 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.ad_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.ag_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.ag_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.g_n
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_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.g_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.am_fm
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_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.am_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.ag_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

−9.877865463999594368227943876935, −9.513145508856414130670156542305, −9.162759996334966976678014034875, −8.653957286510925976905740786362, −8.291946615695808680903976279477, −7.928543513752632560381582392022, −7.24754062810363543825656749735, −7.05968156440363707028324450756, −6.45656483339695475062935820303, −6.15968183716336270652544796730, −5.68008720901731402825000360580, −5.29783298288663241460452518615, −4.83794961968080808395419789308, −4.28288968474340840483283913036, −3.87342366276170379230902321836, −3.28414556135854286180528628589, −2.45428163444717510897293725720, −2.06435114104557519694871934833, −1.57573463571160337084355565395, −0.838348582054009283819936269622, 0.838348582054009283819936269622, 1.57573463571160337084355565395, 2.06435114104557519694871934833, 2.45428163444717510897293725720, 3.28414556135854286180528628589, 3.87342366276170379230902321836, 4.28288968474340840483283913036, 4.83794961968080808395419789308, 5.29783298288663241460452518615, 5.68008720901731402825000360580, 6.15968183716336270652544796730, 6.45656483339695475062935820303, 7.05968156440363707028324450756, 7.24754062810363543825656749735, 7.928543513752632560381582392022, 8.291946615695808680903976279477, 8.653957286510925976905740786362, 9.162759996334966976678014034875, 9.513145508856414130670156542305, 9.877865463999594368227943876935

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