Properties

Label 4-6930e2-1.1-c1e2-0-16
Degree $4$
Conductor $48024900$
Sign $1$
Analytic cond. $3062.10$
Root an. cond. $7.43883$
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 + 3·4-s − 2·5-s − 2·7-s + 4·8-s − 4·10-s − 2·11-s − 4·14-s + 5·16-s − 6·20-s − 4·22-s − 4·23-s + 3·25-s − 6·28-s − 4·29-s + 8·31-s + 6·32-s + 4·35-s − 8·40-s − 4·41-s − 6·44-s − 8·46-s + 3·49-s + 6·50-s − 12·53-s + 4·55-s − 8·56-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s − 0.755·7-s + 1.41·8-s − 1.26·10-s − 0.603·11-s − 1.06·14-s + 5/4·16-s − 1.34·20-s − 0.852·22-s − 0.834·23-s + 3/5·25-s − 1.13·28-s − 0.742·29-s + 1.43·31-s + 1.06·32-s + 0.676·35-s − 1.26·40-s − 0.624·41-s − 0.904·44-s − 1.17·46-s + 3/7·49-s + 0.848·50-s − 1.64·53-s + 0.539·55-s − 1.06·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(48024900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3062.10\)
Root analytic conductor: \(7.43883\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6930} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 48024900,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 24 T + 310 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−7.67748903489183673708373646158, −7.48182660990078748891563576285, −6.83037374498026632908238186793, −6.79937550075434740879568938379, −6.18747179445599159935963859434, −6.15030972567659718751926779198, −5.54986991649816838480846130667, −5.36170187812529085077880780259, −4.77002518917864082764774613405, −4.58460660720671086609781968673, −4.02439405640035891510239979434, −4.00815799229655652624240862658, −3.29323239198889336831080552838, −3.15013242564107021027956354890, −2.60092616688986602321141820903, −2.47555445931060146533630769858, −1.43554775972807461350750042505, −1.39740166594518336368911085882, 0, 0, 1.39740166594518336368911085882, 1.43554775972807461350750042505, 2.47555445931060146533630769858, 2.60092616688986602321141820903, 3.15013242564107021027956354890, 3.29323239198889336831080552838, 4.00815799229655652624240862658, 4.02439405640035891510239979434, 4.58460660720671086609781968673, 4.77002518917864082764774613405, 5.36170187812529085077880780259, 5.54986991649816838480846130667, 6.15030972567659718751926779198, 6.18747179445599159935963859434, 6.79937550075434740879568938379, 6.83037374498026632908238186793, 7.48182660990078748891563576285, 7.67748903489183673708373646158

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