Properties

Label 4-1911e2-1.1-c1e2-0-6
Degree $4$
Conductor $3651921$
Sign $1$
Analytic cond. $232.849$
Root an. cond. $3.90632$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 4-s + 4·6-s + 3·9-s − 4·11-s − 2·12-s + 2·13-s + 16-s − 4·17-s − 6·18-s + 8·22-s − 8·23-s − 2·25-s − 4·26-s − 4·27-s + 4·29-s + 8·31-s + 2·32-s + 8·33-s + 8·34-s + 3·36-s − 4·37-s − 4·39-s − 16·41-s + 8·43-s − 4·44-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s + 9-s − 1.20·11-s − 0.577·12-s + 0.554·13-s + 1/4·16-s − 0.970·17-s − 1.41·18-s + 1.70·22-s − 1.66·23-s − 2/5·25-s − 0.784·26-s − 0.769·27-s + 0.742·29-s + 1.43·31-s + 0.353·32-s + 1.39·33-s + 1.37·34-s + 1/2·36-s − 0.657·37-s − 0.640·39-s − 2.49·41-s + 1.21·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3651921\)    =    \(3^{2} \cdot 7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(232.849\)
Root analytic conductor: \(3.90632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3651921,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 24 T + 314 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−8.958223884801740312196084734694, −8.590789069418742572777387805889, −8.212382684805168410662833884574, −8.097323747454575412305190417297, −7.57970639593912119221298945504, −7.00971164145282174462860886388, −6.76452579565207494874446758602, −6.17229203052613060875050123090, −5.93994469998906887836405329475, −5.56784258456527622396747878423, −4.86777315963195236130899693653, −4.71770319079283481801913599119, −4.06088834140592110241002946061, −3.69469697322211401646804397784, −2.76439236338286257351377342696, −2.44145124664451503766647178108, −1.61542267836316041388675843124, −1.04240775787180465624303775402, 0, 0, 1.04240775787180465624303775402, 1.61542267836316041388675843124, 2.44145124664451503766647178108, 2.76439236338286257351377342696, 3.69469697322211401646804397784, 4.06088834140592110241002946061, 4.71770319079283481801913599119, 4.86777315963195236130899693653, 5.56784258456527622396747878423, 5.93994469998906887836405329475, 6.17229203052613060875050123090, 6.76452579565207494874446758602, 7.00971164145282174462860886388, 7.57970639593912119221298945504, 8.097323747454575412305190417297, 8.212382684805168410662833884574, 8.590789069418742572777387805889, 8.958223884801740312196084734694

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