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Properties

Label 4-2106e2-1.1-c1e2-0-18
Degree $4$
Conductor $4435236$
Sign $1$
Analytic cond. $282.794$
Root an. cond. $4.10079$
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  + 2-s + 2·5-s − 7-s − 8-s + 2·10-s + 11-s + 13-s − 14-s − 16-s + 22-s + 5·23-s + 5·25-s + 26-s − 29-s + 5·31-s − 2·35-s − 6·37-s − 2·40-s + 6·41-s − 43-s + 5·46-s + 4·47-s + 7·49-s + 5·50-s + 6·53-s + 2·55-s + 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.894·5-s − 0.377·7-s − 0.353·8-s + 0.632·10-s + 0.301·11-s + 0.277·13-s − 0.267·14-s − 1/4·16-s + 0.213·22-s + 1.04·23-s + 25-s + 0.196·26-s − 0.185·29-s + 0.898·31-s − 0.338·35-s − 0.986·37-s − 0.316·40-s + 0.937·41-s − 0.152·43-s + 0.737·46-s + 0.583·47-s + 49-s + 0.707·50-s + 0.824·53-s + 0.269·55-s + 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4435236\)    =    \(2^{2} \cdot 3^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(282.794\)
Root analytic conductor: \(4.10079\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4435236,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.372610089\)
\(L(\frac12)\) \(\approx\) \(4.372610089\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
13$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_ab
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.b_ag
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_ak
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 - 5 T + 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.23.af_c
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) 2.29.b_abc
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.31.af_ag
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_af
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$C_2^2$ \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_abf
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.53.ag_el
59$C_2^2$ \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.59.al_ck
61$C_2^2$ \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_abt
67$C_2^2$ \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_ad
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.73.bg_pm
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.89.as_jz
97$C_2^2$ \( 1 - 4 T - 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_add
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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

−9.197207512490866591239490918890, −8.948259500772318192512582167578, −8.649845128395728423932036715924, −8.304125462639533324274680673374, −7.45846254994102234778973421556, −7.41119229562196430011949077610, −6.67491766250058202547537161597, −6.65125063199138435469014726030, −5.94422188044028528590108199423, −5.85560788151075850717964318605, −5.16563845592694640613050217826, −5.09083009267948029210722910228, −4.36653180479692453464114636853, −4.09871211044811972640868527334, −3.42269316978499515452487309320, −3.14775787676207232381687578191, −2.47055649903853158963390266295, −2.16085950916980272299458719544, −1.26413495358641935823218609282, −0.71901979945802806461881257353, 0.71901979945802806461881257353, 1.26413495358641935823218609282, 2.16085950916980272299458719544, 2.47055649903853158963390266295, 3.14775787676207232381687578191, 3.42269316978499515452487309320, 4.09871211044811972640868527334, 4.36653180479692453464114636853, 5.09083009267948029210722910228, 5.16563845592694640613050217826, 5.85560788151075850717964318605, 5.94422188044028528590108199423, 6.65125063199138435469014726030, 6.67491766250058202547537161597, 7.41119229562196430011949077610, 7.45846254994102234778973421556, 8.304125462639533324274680673374, 8.649845128395728423932036715924, 8.948259500772318192512582167578, 9.197207512490866591239490918890

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