Properties

Label 4-1400e2-1.1-c0e2-0-4
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $0.488169$
Root an. cond. $0.835877$
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-s + 7-s − 8-s + 9-s + 14-s − 16-s − 3·17-s + 18-s + 23-s + 3·31-s − 3·34-s + 46-s − 3·47-s − 56-s + 3·62-s + 63-s + 64-s + 2·71-s − 72-s − 79-s + 3·89-s − 3·94-s + 3·103-s − 112-s + 2·113-s − 3·119-s − 121-s + ⋯
L(s)  = 1  + 2-s + 7-s − 8-s + 9-s + 14-s − 16-s − 3·17-s + 18-s + 23-s + 3·31-s − 3·34-s + 46-s − 3·47-s − 56-s + 3·62-s + 63-s + 64-s + 2·71-s − 72-s − 79-s + 3·89-s − 3·94-s + 3·103-s − 112-s + 2·113-s − 3·119-s − 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.488169\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.762852697\)
\(L(\frac12)\) \(\approx\) \(1.762852697\)
\(L(1)\) 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)$
bad2$C_2$ \( 1 - T + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 - T + T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
19$C_2^2$ \( 1 - T^{2} + T^{4} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2^2$ \( 1 - T^{2} + T^{4} \)
61$C_2^2$ \( 1 - T^{2} + T^{4} \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−9.819892793658799178902341034078, −9.755735906599339843976681354806, −8.943008756550614501453968928576, −8.806630611356733707179497744373, −8.396026262808958145846998167879, −8.040552502300954081511740304015, −7.49644272216237756518809882823, −6.83341114731026566192459680126, −6.65274252191565286689560003681, −6.34757648034035299142123961903, −5.87507935106721267083751221869, −4.90853706078785430990466233289, −4.83766775701998998320063301831, −4.58511237240076559861826811341, −4.33494797215586043126524890271, −3.49245235835361237503992114451, −3.13326258275683580224938435960, −2.22311493512140578088754613390, −2.08030175307023728791051325317, −1.00561290690679019429405129794, 1.00561290690679019429405129794, 2.08030175307023728791051325317, 2.22311493512140578088754613390, 3.13326258275683580224938435960, 3.49245235835361237503992114451, 4.33494797215586043126524890271, 4.58511237240076559861826811341, 4.83766775701998998320063301831, 4.90853706078785430990466233289, 5.87507935106721267083751221869, 6.34757648034035299142123961903, 6.65274252191565286689560003681, 6.83341114731026566192459680126, 7.49644272216237756518809882823, 8.040552502300954081511740304015, 8.396026262808958145846998167879, 8.806630611356733707179497744373, 8.943008756550614501453968928576, 9.755735906599339843976681354806, 9.819892793658799178902341034078

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