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Properties

Label 4-8014e2-1.1-c1e2-0-0
Degree $4$
Conductor $64224196$
Sign $1$
Analytic cond. $4094.99$
Root an. cond. $7.99950$
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 − 2·3-s + 3·4-s + 5-s − 4·6-s − 7-s + 4·8-s + 2·9-s + 2·10-s − 11-s − 6·12-s − 2·13-s − 2·14-s − 2·15-s + 5·16-s − 2·17-s + 4·18-s + 3·19-s + 3·20-s + 2·21-s − 2·22-s + 23-s − 8·24-s − 8·25-s − 4·26-s − 6·27-s − 3·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.447·5-s − 1.63·6-s − 0.377·7-s + 1.41·8-s + 2/3·9-s + 0.632·10-s − 0.301·11-s − 1.73·12-s − 0.554·13-s − 0.534·14-s − 0.516·15-s + 5/4·16-s − 0.485·17-s + 0.942·18-s + 0.688·19-s + 0.670·20-s + 0.436·21-s − 0.426·22-s + 0.208·23-s − 1.63·24-s − 8/5·25-s − 0.784·26-s − 1.15·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64224196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64224196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(64224196\)    =    \(2^{2} \cdot 4007^{2}\)
Sign: $1$
Analytic conductor: \(4094.99\)
Root analytic conductor: \(7.99950\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 64224196,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.176188053\)
\(L(\frac12)\) \(\approx\) \(6.176188053\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
4007 \( 1+O(T) \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 13 T + 117 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 122 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - T + 133 T^{2} - p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 3 T + 137 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 7 T + 167 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 9 T + 197 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 30 T + 414 T^{2} + 30 p T^{3} + p^{2} T^{4} \)
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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

−7.71695758525646877009925760569, −7.39254679496680201076473480847, −7.34579833034205860439771828045, −6.81464225209008002508330518478, −6.29165353314662805020702727017, −6.25934494338249563622065640976, −5.77906856840904420163906966088, −5.66263196997954025301302338780, −5.20696056166213764958979489255, −5.07367197985640799798212931818, −4.39686132902433672769485397881, −4.13216134321851127178589350015, −3.97670097723396637355230526489, −3.50459718221109477735529385788, −2.69586966168116995818899767682, −2.46729845883542613944601893479, −2.39468523254376131936127048462, −1.63303017502553530349473868666, −0.847437092911885731410319836871, −0.61818619957471219832621734723, 0.61818619957471219832621734723, 0.847437092911885731410319836871, 1.63303017502553530349473868666, 2.39468523254376131936127048462, 2.46729845883542613944601893479, 2.69586966168116995818899767682, 3.50459718221109477735529385788, 3.97670097723396637355230526489, 4.13216134321851127178589350015, 4.39686132902433672769485397881, 5.07367197985640799798212931818, 5.20696056166213764958979489255, 5.66263196997954025301302338780, 5.77906856840904420163906966088, 6.25934494338249563622065640976, 6.29165353314662805020702727017, 6.81464225209008002508330518478, 7.34579833034205860439771828045, 7.39254679496680201076473480847, 7.71695758525646877009925760569

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