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Properties

Label 4-2376e2-1.1-c1e2-0-7
Degree $4$
Conductor $5645376$
Sign $1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 11-s − 2·13-s − 12·17-s + 12·19-s − 7·23-s + 5·25-s + 10·29-s + 4·37-s + 12·41-s + 8·43-s + 47-s + 7·49-s + 18·53-s − 55-s − 8·61-s − 2·65-s − 13·67-s − 4·73-s + 14·79-s + 6·83-s − 12·85-s + 26·89-s + 12·95-s − 10·97-s − 6·101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.301·11-s − 0.554·13-s − 2.91·17-s + 2.75·19-s − 1.45·23-s + 25-s + 1.85·29-s + 0.657·37-s + 1.87·41-s + 1.21·43-s + 0.145·47-s + 49-s + 2.47·53-s − 0.134·55-s − 1.02·61-s − 0.248·65-s − 1.58·67-s − 0.468·73-s + 1.57·79-s + 0.658·83-s − 1.30·85-s + 2.75·89-s + 1.23·95-s − 1.01·97-s − 0.597·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.695300339\)
\(L(\frac12)\) \(\approx\) \(2.695300339\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_ae
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2^2$ \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.23.h_ba
29$C_2^2$ \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.29.ak_ct
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.31.a_abf
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_dz
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.ai_v
47$C_2^2$ \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) 2.47.ab_abu
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_d
67$C_2^2$ \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.67.n_dy
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.79.ao_en
83$C_2^2$ \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_abv
89$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.89.aba_nj
97$C_2^2$ \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.97.k_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

−9.137453281004785815587701126634, −8.923665497516118641297585453029, −8.424110225154385623216658514168, −8.045115157996152533826734686322, −7.41042953617799825676449204717, −7.38806670701974831035492701545, −6.90396864195286646832731457995, −6.45103959340820889210844749968, −5.96101061516007858782795754759, −5.78840495681244345299561421937, −5.18641532741203251628010489410, −4.77223908191217599324203698637, −4.29007653973748002774471737284, −4.20611728668552518560263234701, −3.31278214715635285807320427463, −2.82916760170696193335109462407, −2.29388214693793777646647371580, −2.23120788569265401248427978338, −1.09514316245703514005357124702, −0.63824393014040962981170178140, 0.63824393014040962981170178140, 1.09514316245703514005357124702, 2.23120788569265401248427978338, 2.29388214693793777646647371580, 2.82916760170696193335109462407, 3.31278214715635285807320427463, 4.20611728668552518560263234701, 4.29007653973748002774471737284, 4.77223908191217599324203698637, 5.18641532741203251628010489410, 5.78840495681244345299561421937, 5.96101061516007858782795754759, 6.45103959340820889210844749968, 6.90396864195286646832731457995, 7.38806670701974831035492701545, 7.41042953617799825676449204717, 8.045115157996152533826734686322, 8.424110225154385623216658514168, 8.923665497516118641297585453029, 9.137453281004785815587701126634

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