Properties

Label 4-96e4-1.1-c1e2-0-3
Degree $4$
Conductor $84934656$
Sign $1$
Analytic cond. $5415.50$
Root an. cond. $8.57846$
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  − 8·7-s + 12·17-s + 16·23-s − 8·25-s − 8·31-s + 4·41-s − 16·47-s + 34·49-s + 20·73-s − 24·79-s − 32·89-s + 16·97-s + 8·103-s + 32·113-s − 96·119-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 128·161-s + 163-s + 167-s − 8·169-s + ⋯
L(s)  = 1  − 3.02·7-s + 2.91·17-s + 3.33·23-s − 8/5·25-s − 1.43·31-s + 0.624·41-s − 2.33·47-s + 34/7·49-s + 2.34·73-s − 2.70·79-s − 3.39·89-s + 1.62·97-s + 0.788·103-s + 3.01·113-s − 8.80·119-s + 0.909·121-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 − 10.0·161-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84934656\)    =    \(2^{20} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(5415.50\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 84934656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.744626289\)
\(L(\frac12)\) \(\approx\) \(1.744626289\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−7.62737769036130342603095132388, −7.45922760667776023734989004425, −7.21039796904218709432625250389, −6.92257730599314885301537395694, −6.45108007309252564268925586539, −6.27240026633574292889261636057, −5.82372132680696849189412715924, −5.58019900362253878027437561144, −5.23724651396779942505594140063, −4.97592687656628299042940663907, −4.21206537344979041421251926597, −3.87558179656833226187493617212, −3.42688180940518004460028982928, −3.14196602028326073410732398546, −3.09409680275504381007938105393, −2.78126539175672080976280028624, −1.91720619574017406770510762004, −1.42476183201926183740417307016, −0.77523009812156767511315706229, −0.41262008320220654175248379335, 0.41262008320220654175248379335, 0.77523009812156767511315706229, 1.42476183201926183740417307016, 1.91720619574017406770510762004, 2.78126539175672080976280028624, 3.09409680275504381007938105393, 3.14196602028326073410732398546, 3.42688180940518004460028982928, 3.87558179656833226187493617212, 4.21206537344979041421251926597, 4.97592687656628299042940663907, 5.23724651396779942505594140063, 5.58019900362253878027437561144, 5.82372132680696849189412715924, 6.27240026633574292889261636057, 6.45108007309252564268925586539, 6.92257730599314885301537395694, 7.21039796904218709432625250389, 7.45922760667776023734989004425, 7.62737769036130342603095132388

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