↑

Properties

Label 4-5850e2-1.1-c1e2-0-43
Degree $4$
Conductor $34222500$
Sign $1$
Analytic cond. $2182.05$
Root an. cond. $6.83465$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 2·7-s + 4·8-s + 2·13-s + 4·14-s + 5·16-s − 10·17-s − 10·19-s − 10·23-s + 4·26-s + 6·28-s − 6·29-s − 8·31-s + 6·32-s − 20·34-s − 4·37-s − 20·38-s − 16·41-s + 4·43-s − 20·46-s − 8·47-s − 6·49-s + 6·52-s − 8·53-s + 8·56-s − 12·58-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.755·7-s + 1.41·8-s + 0.554·13-s + 1.06·14-s + 5/4·16-s − 2.42·17-s − 2.29·19-s − 2.08·23-s + 0.784·26-s + 1.13·28-s − 1.11·29-s − 1.43·31-s + 1.06·32-s − 3.42·34-s − 0.657·37-s − 3.24·38-s − 2.49·41-s + 0.609·43-s − 2.94·46-s − 1.16·47-s − 6/7·49-s + 0.832·52-s − 1.09·53-s + 1.06·56-s − 1.57·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 34222500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34222500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(34222500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2182.05\)
Root analytic conductor: \(6.83465\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 34222500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
5 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_k
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.11.a_c
17$D_{4}$ \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.17.k_cc
19$D_{4}$ \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.19.k_cg
23$D_{4}$ \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.23.k_co
29$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_w
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_ac
41$C_4$ \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.41.q_ew
43$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_cs
47$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_be
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_dy
59$C_4$ \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_ek
61$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_bu
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_4$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_ew
73$D_{4}$ \( 1 - 22 T + 262 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.73.aw_kc
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_dy
89$D_{4}$ \( 1 - 8 T + 174 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.89.ai_gs
97$D_{4}$ \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_gc
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.81122735492602141585430979018, −7.61383602082054390349132229286, −6.90307643933691920894558790339, −6.79015900919429221608068355790, −6.34191398845210435619100740480, −6.25026473738805906660265384068, −5.80670627044348311115959358704, −5.29071582008442050734130698629, −4.93033789994839879741999634505, −4.66019785843464519839118536704, −4.24529391010141017319634547131, −3.95680711280824626523750376584, −3.58310223646203193591377475995, −3.29715971240276180006829501449, −2.29395518141471516794836985667, −2.23689570976657108643725636678, −1.75756537985335038055929929512, −1.64697555635037653880263001564, 0, 0, 1.64697555635037653880263001564, 1.75756537985335038055929929512, 2.23689570976657108643725636678, 2.29395518141471516794836985667, 3.29715971240276180006829501449, 3.58310223646203193591377475995, 3.95680711280824626523750376584, 4.24529391010141017319634547131, 4.66019785843464519839118536704, 4.93033789994839879741999634505, 5.29071582008442050734130698629, 5.80670627044348311115959358704, 6.25026473738805906660265384068, 6.34191398845210435619100740480, 6.79015900919429221608068355790, 6.90307643933691920894558790339, 7.61383602082054390349132229286, 7.81122735492602141585430979018

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