Properties

Label 4-4368e2-1.1-c1e2-0-13
Degree $4$
Conductor $19079424$
Sign $1$
Analytic cond. $1216.52$
Root an. cond. $5.90581$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s + 4·13-s − 4·17-s + 12·23-s + 6·25-s − 4·27-s − 8·39-s − 8·43-s − 49-s + 8·51-s + 8·53-s + 24·61-s − 24·69-s − 12·75-s + 5·81-s + 4·101-s − 28·103-s + 24·107-s + 28·113-s + 12·117-s + 22·121-s + 127-s + 16·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 1.10·13-s − 0.970·17-s + 2.50·23-s + 6/5·25-s − 0.769·27-s − 1.28·39-s − 1.21·43-s − 1/7·49-s + 1.12·51-s + 1.09·53-s + 3.07·61-s − 2.88·69-s − 1.38·75-s + 5/9·81-s + 0.398·101-s − 2.75·103-s + 2.32·107-s + 2.63·113-s + 1.10·117-s + 2·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19079424\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1216.52\)
Root analytic conductor: \(5.90581\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4368} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 19079424,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.412563447\)
\(L(\frac12)\) \(\approx\) \(2.412563447\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−8.544881459057269762140229875293, −8.452700606269748405984881845755, −7.63078502487664671856029875241, −7.39419611348063201568599122664, −6.83033136682861410790766180890, −6.73312554614171988020941052096, −6.54389110651510098374254399382, −5.99044672383959295725746431397, −5.49132057618226255564145175110, −5.30241383272797543260989484947, −4.77054409279933922656430640472, −4.71698001294601351326983741525, −3.91137816516967560657369713802, −3.82060605220317077412457362247, −3.03487274458277689145393711938, −2.86142006283351617578994245020, −2.04964569977334677336367017866, −1.59124711584340676922929112787, −0.792612472193496644052679307864, −0.70228106745844854077682498338, 0.70228106745844854077682498338, 0.792612472193496644052679307864, 1.59124711584340676922929112787, 2.04964569977334677336367017866, 2.86142006283351617578994245020, 3.03487274458277689145393711938, 3.82060605220317077412457362247, 3.91137816516967560657369713802, 4.71698001294601351326983741525, 4.77054409279933922656430640472, 5.30241383272797543260989484947, 5.49132057618226255564145175110, 5.99044672383959295725746431397, 6.54389110651510098374254399382, 6.73312554614171988020941052096, 6.83033136682861410790766180890, 7.39419611348063201568599122664, 7.63078502487664671856029875241, 8.452700606269748405984881845755, 8.544881459057269762140229875293

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