Properties

Label 4-7098e2-1.1-c1e2-0-16
Degree $4$
Conductor $50381604$
Sign $1$
Analytic cond. $3212.37$
Root an. cond. $7.52846$
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  + 2·2-s − 2·3-s + 3·4-s − 5-s − 4·6-s + 2·7-s + 4·8-s + 3·9-s − 2·10-s − 11-s − 6·12-s + 4·14-s + 2·15-s + 5·16-s − 8·17-s + 6·18-s − 19-s − 3·20-s − 4·21-s − 2·22-s + 6·23-s − 8·24-s − 5·25-s − 4·27-s + 6·28-s − 4·29-s + 4·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.447·5-s − 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s − 0.632·10-s − 0.301·11-s − 1.73·12-s + 1.06·14-s + 0.516·15-s + 5/4·16-s − 1.94·17-s + 1.41·18-s − 0.229·19-s − 0.670·20-s − 0.872·21-s − 0.426·22-s + 1.25·23-s − 1.63·24-s − 25-s − 0.769·27-s + 1.13·28-s − 0.742·29-s + 0.730·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50381604\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3212.37\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7098} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good5$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 9 T + 90 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 10 T + 142 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T - 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 9 T + 162 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 9 T + 174 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 150 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 11 T + 204 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 10 T + 202 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
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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

−7.39576992358733845219197991413, −7.32194306624623337443532186219, −6.82263761308819513986186743715, −6.74879298502279568039819330919, −6.32344808495821943724595310314, −5.97943218964778306981878203807, −5.46036227218859083512602350990, −5.15147592987947412395598319790, −4.91785793108166490089740037714, −4.80483742208823909613466079059, −4.09809188634619236656263257782, −4.05489920116188140352250128757, −3.42770198213254346411140112078, −3.22969739252261663427991978943, −2.37794513244268088636600016961, −2.17281720287222007118309200925, −1.59075018939489262340828414174, −1.27099453603286907555061954159, 0, 0, 1.27099453603286907555061954159, 1.59075018939489262340828414174, 2.17281720287222007118309200925, 2.37794513244268088636600016961, 3.22969739252261663427991978943, 3.42770198213254346411140112078, 4.05489920116188140352250128757, 4.09809188634619236656263257782, 4.80483742208823909613466079059, 4.91785793108166490089740037714, 5.15147592987947412395598319790, 5.46036227218859083512602350990, 5.97943218964778306981878203807, 6.32344808495821943724595310314, 6.74879298502279568039819330919, 6.82263761308819513986186743715, 7.32194306624623337443532186219, 7.39576992358733845219197991413

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