Properties

Label 4-95e4-1.1-c1e2-0-3
Degree $4$
Conductor $81450625$
Sign $1$
Analytic cond. $5193.36$
Root an. cond. $8.48910$
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  − 3·2-s − 3·3-s + 4·4-s + 9·6-s + 4·7-s − 3·8-s + 2·9-s − 7·11-s − 12·12-s − 10·13-s − 12·14-s + 3·16-s + 12·17-s − 6·18-s − 12·21-s + 21·22-s + 23-s + 9·24-s + 30·26-s + 6·27-s + 16·28-s + 3·29-s + 15·31-s − 6·32-s + 21·33-s − 36·34-s + 8·36-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s + 3.67·6-s + 1.51·7-s − 1.06·8-s + 2/3·9-s − 2.11·11-s − 3.46·12-s − 2.77·13-s − 3.20·14-s + 3/4·16-s + 2.91·17-s − 1.41·18-s − 2.61·21-s + 4.47·22-s + 0.208·23-s + 1.83·24-s + 5.88·26-s + 1.15·27-s + 3.02·28-s + 0.557·29-s + 2.69·31-s − 1.06·32-s + 3.65·33-s − 6.17·34-s + 4/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81450625\)    =    \(5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(5193.36\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 81450625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2932122219\)
\(L(\frac12)\) \(\approx\) \(0.2932122219\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
19 \( 1 \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 7 T + 3 p T^{2} + 7 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - T + 45 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T + 49 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 15 T + 107 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 3 T + 65 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 43 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + T + 75 T^{2} + p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 16 T + 153 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 9 T + 125 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 11 T + 117 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 173 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 133 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T + 163 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−7.972848467038277614207697710298, −7.74504273822157662521518960545, −7.38672295160848978584028566035, −7.25896755655266951667679659975, −6.58414004613900478741574052913, −6.25426324543796476160813314911, −5.73273526030359952158059098698, −5.38608464899919650827109223317, −5.34562781775155113221910655341, −4.94197422585791770343915639731, −4.65737892637451429163093685866, −4.48766535873733415080207905497, −3.40780357621589447909614807523, −2.96811785830308954929157496838, −2.61595613796707192788643105114, −2.36818751480686467142562514474, −1.56272071381233982663400079465, −1.15708265409167384381239846047, −0.56479220947606063370992410294, −0.41186242450916430239020999668, 0.41186242450916430239020999668, 0.56479220947606063370992410294, 1.15708265409167384381239846047, 1.56272071381233982663400079465, 2.36818751480686467142562514474, 2.61595613796707192788643105114, 2.96811785830308954929157496838, 3.40780357621589447909614807523, 4.48766535873733415080207905497, 4.65737892637451429163093685866, 4.94197422585791770343915639731, 5.34562781775155113221910655341, 5.38608464899919650827109223317, 5.73273526030359952158059098698, 6.25426324543796476160813314911, 6.58414004613900478741574052913, 7.25896755655266951667679659975, 7.38672295160848978584028566035, 7.74504273822157662521518960545, 7.972848467038277614207697710298

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