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Properties

Label 4-209952-1.1-c1e2-0-9
Degree $4$
Conductor $209952$
Sign $1$
Analytic cond. $13.3867$
Root an. cond. $1.91279$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 6·11-s + 7·13-s + 16-s + 6·22-s + 3·23-s + 2·25-s + 7·26-s + 32-s − 17·37-s + 6·44-s + 3·46-s + 6·47-s − 4·49-s + 2·50-s + 7·52-s + 3·59-s − 5·61-s + 64-s − 9·71-s − 8·73-s − 17·74-s + 6·83-s + 6·88-s + 3·92-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.80·11-s + 1.94·13-s + 1/4·16-s + 1.27·22-s + 0.625·23-s + 2/5·25-s + 1.37·26-s + 0.176·32-s − 2.79·37-s + 0.904·44-s + 0.442·46-s + 0.875·47-s − 4/7·49-s + 0.282·50-s + 0.970·52-s + 0.390·59-s − 0.640·61-s + 1/8·64-s − 1.06·71-s − 0.936·73-s − 1.97·74-s + 0.658·83-s + 0.639·88-s + 0.312·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(209952\)    =    \(2^{5} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(13.3867\)
Root analytic conductor: \(1.91279\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 209952,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.393626347\)
\(L(\frac12)\) \(\approx\) \(3.393626347\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ag_w
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.ah_bk
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.19.a_h
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.ad_bc
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.29.a_bu
31$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.31.a_abs
37$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.r_fo
41$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.41.a_abj
43$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.43.a_abp
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ag_dq
53$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.53.a_ca
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.ad_k
61$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.f_cu
67$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.67.a_de
71$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.j_ge
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.i_s
79$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \) 2.79.a_ei
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.83.ag_gk
89$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.89.a_af
97$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.u_if
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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

−8.960865110208532572073026872103, −8.637295360903926283760638534109, −8.294906559543423058875466172457, −7.41336714477434261172621586277, −7.01614033249322101532911667771, −6.49670626739883749240087208333, −6.27421406868778568230548141843, −5.60303328263850849400202950398, −5.15280477815619461574815286605, −4.36158610026643344805249204285, −3.88580106778109219278090941820, −3.51160690888276001095495063565, −2.88930616146844710468768697881, −1.67387743585758454215397662536, −1.26022754237539906715805013491, 1.26022754237539906715805013491, 1.67387743585758454215397662536, 2.88930616146844710468768697881, 3.51160690888276001095495063565, 3.88580106778109219278090941820, 4.36158610026643344805249204285, 5.15280477815619461574815286605, 5.60303328263850849400202950398, 6.27421406868778568230548141843, 6.49670626739883749240087208333, 7.01614033249322101532911667771, 7.41336714477434261172621586277, 8.294906559543423058875466172457, 8.637295360903926283760638534109, 8.960865110208532572073026872103

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