Properties

Label 4-775e2-1.1-c0e2-0-1
Degree $4$
Conductor $600625$
Sign $1$
Analytic cond. $0.149595$
Root an. cond. $0.621912$
Motivic weight $0$
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  + 4-s + 2·9-s − 2·19-s − 2·31-s + 2·36-s − 2·41-s + 49-s + 2·59-s − 64-s + 2·71-s − 2·76-s + 3·81-s + 2·101-s − 2·109-s + 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s + 2·169-s + ⋯
L(s)  = 1  + 4-s + 2·9-s − 2·19-s − 2·31-s + 2·36-s − 2·41-s + 49-s + 2·59-s − 64-s + 2·71-s − 2·76-s + 3·81-s + 2·101-s − 2·109-s + 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s + 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(600625\)    =    \(5^{4} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(0.149595\)
Root analytic conductor: \(0.621912\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{775} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 600625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.226635758\)
\(L(\frac12)\) \(\approx\) \(1.226635758\)
\(L(1)\) 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)$
bad5 \( 1 \)
31$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
7$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2$ \( ( 1 + T + T^{2} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2$ \( ( 1 + T + T^{2} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2$ \( ( 1 - T + T^{2} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2^2$ \( 1 - T^{2} + T^{4} \)
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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

−10.60198222553491738499943501981, −10.48296094574690901419543687142, −9.936768971150411729091678009578, −9.623509709002433401868198565158, −8.935659138547500208168029283980, −8.697667552476901436637523884576, −8.124827019566218910674591906819, −7.40180487878064152861123052266, −7.39319484456533031912666756805, −6.66724582494043593124821260800, −6.63987312362749575996215691703, −6.13037179539428549648806427794, −5.21805494329496496428835159761, −5.04714880886659193247391087163, −4.13136210575692824492621943906, −3.98666071039928900341649711891, −3.39112788915764216126170731891, −2.19562916259559998164863113647, −2.17650365835593031561178200114, −1.39342324416269898907440800239, 1.39342324416269898907440800239, 2.17650365835593031561178200114, 2.19562916259559998164863113647, 3.39112788915764216126170731891, 3.98666071039928900341649711891, 4.13136210575692824492621943906, 5.04714880886659193247391087163, 5.21805494329496496428835159761, 6.13037179539428549648806427794, 6.63987312362749575996215691703, 6.66724582494043593124821260800, 7.39319484456533031912666756805, 7.40180487878064152861123052266, 8.124827019566218910674591906819, 8.697667552476901436637523884576, 8.935659138547500208168029283980, 9.623509709002433401868198565158, 9.936768971150411729091678009578, 10.48296094574690901419543687142, 10.60198222553491738499943501981

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