Properties

Label 4-1323e2-1.1-c1e2-0-7
Degree $4$
Conductor $1750329$
Sign $1$
Analytic cond. $111.602$
Root an. cond. $3.25026$
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  − 2·2-s − 4-s + 5-s + 8·8-s − 2·10-s + 5·11-s − 5·13-s − 7·16-s − 3·17-s + 19-s − 20-s − 10·22-s + 3·23-s + 5·25-s + 10·26-s − 29-s − 14·32-s + 6·34-s − 3·37-s − 2·38-s + 8·40-s + 5·41-s + 43-s − 5·44-s − 6·46-s − 10·50-s + 5·52-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 0.447·5-s + 2.82·8-s − 0.632·10-s + 1.50·11-s − 1.38·13-s − 7/4·16-s − 0.727·17-s + 0.229·19-s − 0.223·20-s − 2.13·22-s + 0.625·23-s + 25-s + 1.96·26-s − 0.185·29-s − 2.47·32-s + 1.02·34-s − 0.493·37-s − 0.324·38-s + 1.26·40-s + 0.780·41-s + 0.152·43-s − 0.753·44-s − 0.884·46-s − 1.41·50-s + 0.693·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1750329\)    =    \(3^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(111.602\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1750329,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9022964901\)
\(L(\frac12)\) \(\approx\) \(0.9022964901\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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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

−9.535338170024283781412128220456, −9.452751642132007660349974921532, −9.072761087674040346065979525254, −8.893537615000344072015385619258, −8.180150274112877491943119789699, −8.134515448776171351797896993779, −7.48441836003620093879257282836, −7.10730281059877771206481189488, −6.58699393019015893816785816125, −6.52643661197486758932776076302, −5.34319958937089259603435096710, −5.30896409065956402084594706523, −4.70029101271739953483134895144, −4.43129392912592130279411663122, −3.70992631059854340631228662438, −3.44407408799182771456716684405, −2.16190965338140163366562467892, −2.09978212722409360862569811088, −0.980369902927552590734704936391, −0.68306832566709173924618476460, 0.68306832566709173924618476460, 0.980369902927552590734704936391, 2.09978212722409360862569811088, 2.16190965338140163366562467892, 3.44407408799182771456716684405, 3.70992631059854340631228662438, 4.43129392912592130279411663122, 4.70029101271739953483134895144, 5.30896409065956402084594706523, 5.34319958937089259603435096710, 6.52643661197486758932776076302, 6.58699393019015893816785816125, 7.10730281059877771206481189488, 7.48441836003620093879257282836, 8.134515448776171351797896993779, 8.180150274112877491943119789699, 8.893537615000344072015385619258, 9.072761087674040346065979525254, 9.452751642132007660349974921532, 9.535338170024283781412128220456

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