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Properties

Label 4-855e2-1.1-c0e2-0-2
Degree $4$
Conductor $731025$
Sign $1$
Analytic cond. $0.182073$
Root an. cond. $0.653223$
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  + 2·5-s − 16-s − 2·19-s + 3·25-s + 2·49-s − 2·80-s − 4·95-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 2·5-s − 16-s − 2·19-s + 3·25-s + 2·49-s − 2·80-s − 4·95-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(731025\)    =    \(3^{4} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.182073\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 731025,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.257104105\)
\(L(\frac12)\) \(\approx\) \(1.257104105\)
\(L(1)\) 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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_2^2$ \( 1 + T^{4} \)
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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

−10.45792011474811071120776346473, −10.45086148821719386969462945688, −9.623923462398388958674827630161, −9.406427590787638692703192745680, −8.855992603957737578040280210684, −8.770249463260139601258757786070, −8.234173153575635250964562267969, −7.53874129702747483862315240467, −7.01667909629407094397518069355, −6.54735645376233508662178637077, −6.33609500404421226072248267111, −5.88672294809600988220141059621, −5.31776636594026645937298083103, −4.98490716495764179214499644337, −4.31432945832676453282242500406, −3.93930107019701489209123059048, −2.94017852514384129976399413879, −2.35713804217991572148981181520, −2.13367658955366694371033754489, −1.30458727967014322993710946472, 1.30458727967014322993710946472, 2.13367658955366694371033754489, 2.35713804217991572148981181520, 2.94017852514384129976399413879, 3.93930107019701489209123059048, 4.31432945832676453282242500406, 4.98490716495764179214499644337, 5.31776636594026645937298083103, 5.88672294809600988220141059621, 6.33609500404421226072248267111, 6.54735645376233508662178637077, 7.01667909629407094397518069355, 7.53874129702747483862315240467, 8.234173153575635250964562267969, 8.770249463260139601258757786070, 8.855992603957737578040280210684, 9.406427590787638692703192745680, 9.623923462398388958674827630161, 10.45086148821719386969462945688, 10.45792011474811071120776346473

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