Properties

Label 4-666e2-1.1-c1e2-0-4
Degree $4$
Conductor $443556$
Sign $1$
Analytic cond. $28.2815$
Root an. cond. $2.30608$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 5·7-s + 6·11-s + 16-s + 2·25-s − 5·28-s − 7·37-s + 3·41-s + 6·44-s + 6·47-s + 7·49-s + 64-s + 7·67-s − 9·71-s + 13·73-s − 30·77-s + 24·83-s + 2·100-s + 15·101-s − 5·112-s + 14·121-s + 127-s + 131-s + 137-s + 139-s − 7·148-s + 149-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.88·7-s + 1.80·11-s + 1/4·16-s + 2/5·25-s − 0.944·28-s − 1.15·37-s + 0.468·41-s + 0.904·44-s + 0.875·47-s + 49-s + 1/8·64-s + 0.855·67-s − 1.06·71-s + 1.52·73-s − 3.41·77-s + 2.63·83-s + 1/5·100-s + 1.49·101-s − 0.472·112-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.575·148-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(443556\)    =    \(2^{2} \cdot 3^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(28.2815\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 443556,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.752329225\)
\(L(\frac12)\) \(\approx\) \(1.752329225\)
\(L(\frac{3}{2})\) 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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
37$C_2$ \( 1 + 7 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
43$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 9 T + p T^{2} ) \)
89$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
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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

−8.820569452891575312035096497533, −8.139743426164762426896802064442, −7.51577421793174998945463067678, −7.08435344514735874699303383882, −6.61926369665901221466870868365, −6.36198654373467615234893955818, −6.10486610480185424206658058416, −5.36588369481349339860294761879, −4.78479566635529712614734648898, −3.90006589527511075765360779631, −3.70685499028521781137005262636, −3.17349333332914601855466930718, −2.51293610138926031796492183564, −1.69688687831240788452505004580, −0.72582147763788633757709097494, 0.72582147763788633757709097494, 1.69688687831240788452505004580, 2.51293610138926031796492183564, 3.17349333332914601855466930718, 3.70685499028521781137005262636, 3.90006589527511075765360779631, 4.78479566635529712614734648898, 5.36588369481349339860294761879, 6.10486610480185424206658058416, 6.36198654373467615234893955818, 6.61926369665901221466870868365, 7.08435344514735874699303383882, 7.51577421793174998945463067678, 8.139743426164762426896802064442, 8.820569452891575312035096497533

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