Properties

Label 4-28e4-1.1-c1e2-0-34
Degree $4$
Conductor $614656$
Sign $1$
Analytic cond. $39.1909$
Root an. cond. $2.50205$
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  + 3·9-s + 4·11-s + 8·23-s + 5·25-s + 4·29-s + 6·37-s + 24·43-s + 10·53-s + 4·67-s − 32·71-s + 8·79-s + 12·99-s − 20·107-s − 18·109-s + 4·113-s + 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + ⋯
L(s)  = 1  + 9-s + 1.20·11-s + 1.66·23-s + 25-s + 0.742·29-s + 0.986·37-s + 3.65·43-s + 1.37·53-s + 0.488·67-s − 3.79·71-s + 0.900·79-s + 1.20·99-s − 1.93·107-s − 1.72·109-s + 0.376·113-s + 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 − 2·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(614656\)    =    \(2^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(39.1909\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 614656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.803270613\)
\(L(\frac12)\) \(\approx\) \(2.803270613\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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

−10.38187217000862441395582993407, −10.25299085848919381388883783340, −9.443538981855195754142310424490, −9.303195903446622637082562170068, −8.861310055164679123751109167298, −8.601496357107932162719582229730, −7.76976371199354797625952189785, −7.47677292364731850174481204849, −6.89261007070198055991557205592, −6.86917580118826050726950422152, −5.99269046729600287746644129766, −5.88597164984551868472163443226, −4.93281723194342072687458241464, −4.69580960775803873789133841995, −3.96240444354551282811832752739, −3.87753582944241538986659644053, −2.70757154290335912035174292591, −2.62577199349411742179891815778, −1.20903961340541875322938612455, −1.12841179400576921434623352143, 1.12841179400576921434623352143, 1.20903961340541875322938612455, 2.62577199349411742179891815778, 2.70757154290335912035174292591, 3.87753582944241538986659644053, 3.96240444354551282811832752739, 4.69580960775803873789133841995, 4.93281723194342072687458241464, 5.88597164984551868472163443226, 5.99269046729600287746644129766, 6.86917580118826050726950422152, 6.89261007070198055991557205592, 7.47677292364731850174481204849, 7.76976371199354797625952189785, 8.601496357107932162719582229730, 8.861310055164679123751109167298, 9.303195903446622637082562170068, 9.443538981855195754142310424490, 10.25299085848919381388883783340, 10.38187217000862441395582993407

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