Properties

Label 4-501120-1.1-c1e2-0-29
Degree $4$
Conductor $501120$
Sign $-1$
Analytic cond. $31.9518$
Root an. cond. $2.37751$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s + 15-s + 16-s + 18-s − 20-s − 8·23-s − 24-s − 2·25-s − 27-s + 7·29-s + 30-s + 32-s + 36-s − 40-s − 45-s − 8·46-s − 48-s + 2·49-s − 2·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.223·20-s − 1.66·23-s − 0.204·24-s − 2/5·25-s − 0.192·27-s + 1.29·29-s + 0.182·30-s + 0.176·32-s + 1/6·36-s − 0.158·40-s − 0.149·45-s − 1.17·46-s − 0.144·48-s + 2/7·49-s − 0.282·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(501120\)    =    \(2^{7} \cdot 3^{3} \cdot 5 \cdot 29\)
Sign: $-1$
Analytic conductor: \(31.9518\)
Root analytic conductor: \(2.37751\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 501120,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
3$C_1$ \( 1 + T \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 6 T + p T^{2} ) \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−8.051908216470238057440330014728, −7.88551805899859417026024801742, −7.32180685334752387551398534620, −6.80067253769893243162394179020, −6.34935527860674297471983930090, −5.92635861416231740748839277363, −5.60906919627425914493351015989, −4.73619671326860617932019969609, −4.62101926168423720356525612494, −4.01225456029838954639178357945, −3.45416308046384759045446148119, −2.85299955888631046531897370771, −2.07684636761464869855794223498, −1.31927290492621886741124002016, 0, 1.31927290492621886741124002016, 2.07684636761464869855794223498, 2.85299955888631046531897370771, 3.45416308046384759045446148119, 4.01225456029838954639178357945, 4.62101926168423720356525612494, 4.73619671326860617932019969609, 5.60906919627425914493351015989, 5.92635861416231740748839277363, 6.34935527860674297471983930090, 6.80067253769893243162394179020, 7.32180685334752387551398534620, 7.88551805899859417026024801742, 8.051908216470238057440330014728

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