Properties

Label 4-4400e2-1.1-c1e2-0-6
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 − 14·31-s − 16·41-s + 10·49-s + 10·59-s + 24·61-s + 6·71-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 + 173-s + 179-s + ⋯
L(s)  = 1  + 5/3·9-s − 0.603·11-s − 2.51·31-s − 2.49·41-s + 10/7·49-s + 1.30·59-s + 3.07·61-s + 0.712·71-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 + 0.0760·173-s + 0.0747·179-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 19360000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.013615322\)
\(L(\frac12)\) \(\approx\) \(2.013615322\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
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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

−8.665698093272711287546339429481, −8.259006030621535316934884903817, −7.81322293939355045824314336652, −7.14782282991405075375376931907, −7.08844497532134828477364697417, −7.00602941825289942251902585716, −6.59101159067835030906321698276, −5.72563659315937116859868993227, −5.51886163905214676100774541635, −5.43151271222498705066696534751, −4.75390618813271717153623928527, −4.40237965223868227791761932714, −4.01409720753106071396164710116, −3.54728046189461839761545605800, −3.38291483460000137498878470416, −2.51655034540659664564190265948, −2.18608893983310120829240641878, −1.65904108881844393081677704161, −1.23830926299751393356431473401, −0.40576330652328546375250683797, 0.40576330652328546375250683797, 1.23830926299751393356431473401, 1.65904108881844393081677704161, 2.18608893983310120829240641878, 2.51655034540659664564190265948, 3.38291483460000137498878470416, 3.54728046189461839761545605800, 4.01409720753106071396164710116, 4.40237965223868227791761932714, 4.75390618813271717153623928527, 5.43151271222498705066696534751, 5.51886163905214676100774541635, 5.72563659315937116859868993227, 6.59101159067835030906321698276, 7.00602941825289942251902585716, 7.08844497532134828477364697417, 7.14782282991405075375376931907, 7.81322293939355045824314336652, 8.259006030621535316934884903817, 8.665698093272711287546339429481

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