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Properties

Label 4-145e2-1.1-c0e2-0-0
Degree $4$
Conductor $21025$
Sign $1$
Analytic cond. $0.00523661$
Root an. cond. $0.269006$
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  − 2·7-s + 2·13-s − 16-s − 2·23-s − 25-s + 2·49-s − 2·53-s + 2·67-s − 81-s + 2·83-s − 4·91-s − 2·103-s + 2·107-s + 2·112-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  − 2·7-s + 2·13-s − 16-s − 2·23-s − 25-s + 2·49-s − 2·53-s + 2·67-s − 81-s + 2·83-s − 4·91-s − 2·103-s + 2·107-s + 2·112-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + 163-s + 167-s + 2·169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21025\)    =    \(5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(0.00523661\)
Root analytic conductor: \(0.269006\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21025,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3606454480\)
\(L(\frac12)\) \(\approx\) \(0.3606454480\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
29$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
3$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−13.34142935627856553675093957670, −13.32993598963205229484444286850, −12.54574876897768236408186826010, −12.28442879818933150612535536041, −11.39422617378668995229584482205, −11.24964052958724875129278183064, −10.41353880598393309285793247204, −10.01740528623592685061557402512, −9.403540390534429671056392011072, −9.180632172722448789636496637107, −8.303183790762553082209274538642, −7.992400538794846214878210654745, −7.01113802604147820072281225945, −6.42492806225623045654874444485, −6.16509560494874622589673106473, −5.63166093877739375271477810465, −4.39077343411487350689405183883, −3.72658814719173165662651788935, −3.27742719664987583238506859129, −2.07354667645135374752323359783, 2.07354667645135374752323359783, 3.27742719664987583238506859129, 3.72658814719173165662651788935, 4.39077343411487350689405183883, 5.63166093877739375271477810465, 6.16509560494874622589673106473, 6.42492806225623045654874444485, 7.01113802604147820072281225945, 7.992400538794846214878210654745, 8.303183790762553082209274538642, 9.180632172722448789636496637107, 9.403540390534429671056392011072, 10.01740528623592685061557402512, 10.41353880598393309285793247204, 11.24964052958724875129278183064, 11.39422617378668995229584482205, 12.28442879818933150612535536041, 12.54574876897768236408186826010, 13.32993598963205229484444286850, 13.34142935627856553675093957670

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