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Properties

Label 4-450e2-1.1-c1e2-0-4
Degree $4$
Conductor $202500$
Sign $1$
Analytic cond. $12.9115$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 6·11-s + 16-s − 10·19-s + 4·31-s + 6·41-s − 6·44-s + 10·49-s + 4·61-s − 64-s − 24·71-s + 10·76-s + 20·79-s + 30·89-s + 36·101-s + 20·109-s + 5·121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 6·164-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.80·11-s + 1/4·16-s − 2.29·19-s + 0.718·31-s + 0.937·41-s − 0.904·44-s + 10/7·49-s + 0.512·61-s − 1/8·64-s − 2.84·71-s + 1.14·76-s + 2.25·79-s + 3.17·89-s + 3.58·101-s + 1.91·109-s + 5/11·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s − 0.468·164-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(202500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(12.9115\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 202500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.525812872\)
\(L(\frac12)\) \(\approx\) \(1.525812872\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.17.a_az
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.19.k_cl
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.41.ag_dn
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.67.a_bj
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.73.a_az
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.83.a_adh
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.89.abe_pn
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−11.55559769687808258146981381899, −10.71432999558965687424480725120, −10.36488722718985008200889711497, −10.15627762693614506314734218646, −9.203423758695799293207511199080, −9.101289914422487788296466710397, −8.815797402575775945540480326978, −8.234476345716487980298940491831, −7.69602915479041042990336412423, −7.13740507175189545308739624810, −6.56959845574219540452047617120, −6.06601673927401130057705833198, −5.98474404169795160238665735821, −4.70701752447342561505061485988, −4.70479388462989093193336425987, −3.82711533611488512591428116161, −3.68938504086695471403930522546, −2.54363682198174938006208109249, −1.87207435922589792516046536963, −0.819192104519463078705639082930, 0.819192104519463078705639082930, 1.87207435922589792516046536963, 2.54363682198174938006208109249, 3.68938504086695471403930522546, 3.82711533611488512591428116161, 4.70479388462989093193336425987, 4.70701752447342561505061485988, 5.98474404169795160238665735821, 6.06601673927401130057705833198, 6.56959845574219540452047617120, 7.13740507175189545308739624810, 7.69602915479041042990336412423, 8.234476345716487980298940491831, 8.815797402575775945540480326978, 9.101289914422487788296466710397, 9.203423758695799293207511199080, 10.15627762693614506314734218646, 10.36488722718985008200889711497, 10.71432999558965687424480725120, 11.55559769687808258146981381899

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