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Properties

Label 4-675e2-1.1-c1e2-0-3
Degree $4$
Conductor $455625$
Sign $1$
Analytic cond. $29.0510$
Root an. cond. $2.32161$
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  + 3·4-s − 6·7-s + 4·13-s + 5·16-s + 2·19-s − 18·28-s − 6·31-s + 2·37-s − 2·43-s + 13·49-s + 12·52-s + 14·61-s + 3·64-s − 24·67-s + 22·73-s + 6·76-s + 30·79-s − 24·91-s + 14·97-s + 14·103-s + 34·109-s − 30·112-s + 6·121-s − 18·124-s + 127-s + 131-s − 12·133-s + ⋯
L(s)  = 1  + 3/2·4-s − 2.26·7-s + 1.10·13-s + 5/4·16-s + 0.458·19-s − 3.40·28-s − 1.07·31-s + 0.328·37-s − 0.304·43-s + 13/7·49-s + 1.66·52-s + 1.79·61-s + 3/8·64-s − 2.93·67-s + 2.57·73-s + 0.688·76-s + 3.37·79-s − 2.51·91-s + 1.42·97-s + 1.37·103-s + 3.25·109-s − 2.83·112-s + 6/11·121-s − 1.61·124-s + 0.0887·127-s + 0.0873·131-s − 1.04·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(455625\)    =    \(3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(29.0510\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 455625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.057706977\)
\(L(\frac12)\) \(\approx\) \(2.057706977\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
5 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.17.a_g
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.29.a_be
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.37.ac_cx
41$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.41.a_cc
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.47.a_co
53$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.53.a_da
59$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.59.a_g
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.61.ao_gp
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.71.a_be
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.79.abe_ot
83$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.83.a_adi
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.97.ao_jj
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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

−11.01949562177065143244385887088, −10.26970462124441613773932271582, −9.753131323959798992367574928052, −9.716433023315420714720090309443, −8.854970762158158320656806156993, −8.832137915590806122752744179632, −7.971496139426878862089246138259, −7.35658528880860920168453168465, −7.30537812173058972741068848350, −6.45025569392483847593973729007, −6.35866305503470250365889135222, −6.16482571886967247338423844759, −5.48578857875682619499962455932, −4.86773569835033415573584131310, −3.85601571529966850519457451903, −3.35304967251654141077534059260, −3.30375076762567740142856987476, −2.44210086430761096782746783788, −1.87220090675730271137255508551, −0.74637271727844988252325148508, 0.74637271727844988252325148508, 1.87220090675730271137255508551, 2.44210086430761096782746783788, 3.30375076762567740142856987476, 3.35304967251654141077534059260, 3.85601571529966850519457451903, 4.86773569835033415573584131310, 5.48578857875682619499962455932, 6.16482571886967247338423844759, 6.35866305503470250365889135222, 6.45025569392483847593973729007, 7.30537812173058972741068848350, 7.35658528880860920168453168465, 7.971496139426878862089246138259, 8.832137915590806122752744179632, 8.854970762158158320656806156993, 9.716433023315420714720090309443, 9.753131323959798992367574928052, 10.26970462124441613773932271582, 11.01949562177065143244385887088

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