Properties

Label 4-4400e2-1.1-c1e2-0-10
Degree $4$
Conductor $19360000$
Sign $1$
Analytic cond. $1234.41$
Root an. cond. $5.92740$
Motivic weight $1$
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  + 5·9-s − 2·11-s + 10·19-s − 10·29-s + 6·31-s + 4·41-s + 5·49-s − 20·59-s + 14·61-s − 14·71-s + 20·79-s + 16·81-s + 30·89-s − 10·99-s + 4·101-s + 20·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯
L(s)  = 1  + 5/3·9-s − 0.603·11-s + 2.29·19-s − 1.85·29-s + 1.07·31-s + 0.624·41-s + 5/7·49-s − 2.60·59-s + 1.79·61-s − 1.66·71-s + 2.25·79-s + 16/9·81-s + 3.17·89-s − 1.00·99-s + 0.398·101-s + 1.91·109-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19360000\)    =    \(2^{8} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1234.41\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 19360000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.327037102\)
\(L(\frac12)\) \(\approx\) \(3.327037102\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−8.630115870905812250447134789832, −7.86431262580472197481398636551, −7.67400800422317978086486346328, −7.50388061895654977455833243336, −7.36655432545901952934601046984, −6.77388083954113090036369487563, −6.24675137623678263137202694744, −6.14940308104742102297073672865, −5.49716838571585272561222543762, −5.15672748371605060693335587419, −4.83340720535741041548503002849, −4.55311367612276139804671757300, −3.89095418535510856139225343277, −3.55261454266482824770497547460, −3.36234172967528526257991092007, −2.49770512258210055935742367793, −2.34068969255702141981159566038, −1.48553797125544855704940485807, −1.24008036199597980598032262397, −0.55329352367741145272176896457, 0.55329352367741145272176896457, 1.24008036199597980598032262397, 1.48553797125544855704940485807, 2.34068969255702141981159566038, 2.49770512258210055935742367793, 3.36234172967528526257991092007, 3.55261454266482824770497547460, 3.89095418535510856139225343277, 4.55311367612276139804671757300, 4.83340720535741041548503002849, 5.15672748371605060693335587419, 5.49716838571585272561222543762, 6.14940308104742102297073672865, 6.24675137623678263137202694744, 6.77388083954113090036369487563, 7.36655432545901952934601046984, 7.50388061895654977455833243336, 7.67400800422317978086486346328, 7.86431262580472197481398636551, 8.630115870905812250447134789832

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