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Properties

Label 4-5200e2-1.1-c1e2-0-13
Degree $4$
Conductor $27040000$
Sign $1$
Analytic cond. $1724.09$
Root an. cond. $6.44377$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 6·11-s − 2·13-s + 10·19-s − 6·23-s + 2·27-s − 4·29-s + 6·31-s − 12·33-s − 8·37-s + 4·39-s − 16·41-s − 2·43-s − 16·47-s − 2·49-s − 16·53-s − 20·57-s + 14·59-s − 4·61-s + 4·67-s + 12·69-s − 6·71-s − 12·79-s − 81-s + 16·83-s + 8·87-s + 4·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.80·11-s − 0.554·13-s + 2.29·19-s − 1.25·23-s + 0.384·27-s − 0.742·29-s + 1.07·31-s − 2.08·33-s − 1.31·37-s + 0.640·39-s − 2.49·41-s − 0.304·43-s − 2.33·47-s − 2/7·49-s − 2.19·53-s − 2.64·57-s + 1.82·59-s − 0.512·61-s + 0.488·67-s + 1.44·69-s − 0.712·71-s − 1.35·79-s − 1/9·81-s + 1.75·83-s + 0.857·87-s + 0.423·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27040000\)    =    \(2^{8} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1724.09\)
Root analytic conductor: \(6.44377\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 27040000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_bc
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$D_{4}$ \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.19.ak_ci
23$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_bc
29$D_{4}$ \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_by
31$D_{4}$ \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_cq
37$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_bq
41$D_{4}$ \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.41.q_fe
43$D_{4}$ \( 1 + 2 T + 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_dg
47$D_{4}$ \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.47.q_fq
53$D_{4}$ \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.53.q_gc
59$D_{4}$ \( 1 - 14 T + 140 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_fk
61$D_{4}$ \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_ek
67$D_{4}$ \( 1 - 4 T - 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_acc
71$D_{4}$ \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_cy
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_ha
83$D_{4}$ \( 1 - 16 T + 218 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.83.aq_ik
89$D_{4}$ \( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.89.ae_fe
97$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_g
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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

−7.891252500463834887192062122854, −7.83322038250952797704357135754, −7.06224287384052973310447466368, −6.85779993843081023039767700994, −6.53949829978169963906621921431, −6.40556034482060990413259131756, −5.68258185645459828073964222386, −5.59907380321185223939227979463, −5.13209752501616089528562632728, −4.87161293504776806644727113079, −4.48382599271845081471269316626, −3.78316007533201144157802217709, −3.56720606627189619447794128968, −3.23714926833355999670816145980, −2.69270307217589765066032218328, −1.91906687317289759351177046723, −1.32530472967473519653078788979, −1.32296823242303117147726618718, 0, 0, 1.32296823242303117147726618718, 1.32530472967473519653078788979, 1.91906687317289759351177046723, 2.69270307217589765066032218328, 3.23714926833355999670816145980, 3.56720606627189619447794128968, 3.78316007533201144157802217709, 4.48382599271845081471269316626, 4.87161293504776806644727113079, 5.13209752501616089528562632728, 5.59907380321185223939227979463, 5.68258185645459828073964222386, 6.40556034482060990413259131756, 6.53949829978169963906621921431, 6.85779993843081023039767700994, 7.06224287384052973310447466368, 7.83322038250952797704357135754, 7.891252500463834887192062122854

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