Properties

Label 4-60500-1.1-c1e2-0-4
Degree $4$
Conductor $60500$
Sign $1$
Analytic cond. $3.85753$
Root an. cond. $1.40144$
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  + 4-s + 5-s − 5·9-s + 2·11-s + 16-s + 10·19-s + 20-s + 25-s + 10·29-s − 6·31-s − 5·36-s + 4·41-s + 2·44-s − 5·45-s − 5·49-s + 2·55-s − 20·59-s + 14·61-s + 64-s + 14·71-s + 10·76-s + 20·79-s + 80-s + 16·81-s − 30·89-s + 10·95-s − 10·99-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.447·5-s − 5/3·9-s + 0.603·11-s + 1/4·16-s + 2.29·19-s + 0.223·20-s + 1/5·25-s + 1.85·29-s − 1.07·31-s − 5/6·36-s + 0.624·41-s + 0.301·44-s − 0.745·45-s − 5/7·49-s + 0.269·55-s − 2.60·59-s + 1.79·61-s + 1/8·64-s + 1.66·71-s + 1.14·76-s + 2.25·79-s + 0.111·80-s + 16/9·81-s − 3.17·89-s + 1.02·95-s − 1.00·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(60500\)    =    \(2^{2} \cdot 5^{3} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3.85753\)
Root analytic conductor: \(1.40144\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 60500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.653218027\)
\(L(\frac12)\) \(\approx\) \(1.653218027\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( 1 - T \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.a_ap
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.19.ak_cl
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.29.ak_df
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.37.a_cn
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.53.a_eb
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.59.u_ik
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.61.ao_gp
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.71.ao_hj
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.a_aby
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.89.be_pn
97$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.a_by
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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

−9.696869758195084295021589593937, −9.537545307560587613009226803801, −9.051313158030191365780373658814, −8.319735201663290337491520314526, −8.040279775818558509679448913672, −7.35838560113530362823348866724, −6.71133528252224918895941305968, −6.32452856749704006618306134382, −5.51294812234663797155432190910, −5.43280032020194587138281999658, −4.57429547993073909832853594043, −3.46996281594193181675719351221, −3.07694177217990984214438626883, −2.34744313488402780667751356221, −1.14442070453994046878747912184, 1.14442070453994046878747912184, 2.34744313488402780667751356221, 3.07694177217990984214438626883, 3.46996281594193181675719351221, 4.57429547993073909832853594043, 5.43280032020194587138281999658, 5.51294812234663797155432190910, 6.32452856749704006618306134382, 6.71133528252224918895941305968, 7.35838560113530362823348866724, 8.040279775818558509679448913672, 8.319735201663290337491520314526, 9.051313158030191365780373658814, 9.537545307560587613009226803801, 9.696869758195084295021589593937

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