Properties

Label 4-3675e2-1.1-c1e2-0-7
Degree $4$
Conductor $13505625$
Sign $1$
Analytic cond. $861.130$
Root an. cond. $5.41710$
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  + 2-s − 2·3-s − 2·4-s − 2·6-s − 3·8-s + 3·9-s + 4·12-s + 4·13-s + 16-s + 4·17-s + 3·18-s + 4·19-s + 4·23-s + 6·24-s + 4·26-s − 4·27-s − 6·29-s + 8·31-s + 2·32-s + 4·34-s − 6·36-s + 2·37-s + 4·38-s − 8·39-s − 4·43-s + 4·46-s + 12·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 4-s − 0.816·6-s − 1.06·8-s + 9-s + 1.15·12-s + 1.10·13-s + 1/4·16-s + 0.970·17-s + 0.707·18-s + 0.917·19-s + 0.834·23-s + 1.22·24-s + 0.784·26-s − 0.769·27-s − 1.11·29-s + 1.43·31-s + 0.353·32-s + 0.685·34-s − 36-s + 0.328·37-s + 0.648·38-s − 1.28·39-s − 0.609·43-s + 0.589·46-s + 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13505625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(861.130\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3675} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13505625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.432056105\)
\(L(\frac12)\) \(\approx\) \(2.432056105\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
7 \( 1 \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 2 T + 55 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 12 T + 125 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 173 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 20 T + 226 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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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

−8.716424920937127424608902795774, −8.501951421340583370945141673758, −7.80032570571314572026789764339, −7.62825275732571359821152060969, −7.23713402589484232943368699180, −6.70647755003476607030460074262, −6.29083246312614561869923438872, −6.03403930986436437592101397190, −5.58518575015332333207613494251, −5.30852168681852901648024062341, −4.98624811114539665539354289203, −4.67505048655027284685722919623, −4.07331552786894921182291248463, −3.91037351924812594223985792086, −3.27086315745740890875196620470, −3.16537886852318153350254269490, −2.24724952467315039669351320518, −1.57050684715428305011054075461, −0.77847757740791034949696327729, −0.71075795450917434645347341932, 0.71075795450917434645347341932, 0.77847757740791034949696327729, 1.57050684715428305011054075461, 2.24724952467315039669351320518, 3.16537886852318153350254269490, 3.27086315745740890875196620470, 3.91037351924812594223985792086, 4.07331552786894921182291248463, 4.67505048655027284685722919623, 4.98624811114539665539354289203, 5.30852168681852901648024062341, 5.58518575015332333207613494251, 6.03403930986436437592101397190, 6.29083246312614561869923438872, 6.70647755003476607030460074262, 7.23713402589484232943368699180, 7.62825275732571359821152060969, 7.80032570571314572026789764339, 8.501951421340583370945141673758, 8.716424920937127424608902795774

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