Properties

Label 4-1216e2-1.1-c1e2-0-13
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
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  − 6·7-s + 5·9-s + 6·17-s + 18·23-s + 10·25-s − 12·31-s + 12·41-s + 13·49-s − 30·63-s + 22·73-s + 24·79-s + 16·81-s − 16·97-s − 12·103-s − 24·113-s − 36·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 30·153-s + 157-s − 108·161-s + 163-s + ⋯
L(s)  = 1  − 2.26·7-s + 5/3·9-s + 1.45·17-s + 3.75·23-s + 2·25-s − 2.15·31-s + 1.87·41-s + 13/7·49-s − 3.77·63-s + 2.57·73-s + 2.70·79-s + 16/9·81-s − 1.62·97-s − 1.18·103-s − 2.25·113-s − 3.30·119-s + 2·121-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 + 2.42·153-s + 0.0798·157-s − 8.51·161-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.433738716\)
\(L(\frac12)\) \(\approx\) \(2.433738716\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
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

−9.661165550558986193511889780429, −9.452082351369450189442281696885, −9.338297278609692927807365341382, −9.064455935931297151545476565667, −8.312700185860270809450382307879, −7.77307587699736122115391753624, −7.16052590256910063539794378756, −7.08070584991519485848520445672, −6.69205073326629599870617058586, −6.50250727412608230979435762145, −5.55270838919019823822788404583, −5.43690883218433792305781071075, −4.79260310657974610597233093823, −4.35336587019880043428892678064, −3.46764598695459551704989318974, −3.39567998387535460812683799868, −3.01000527337709967171374865417, −2.24773209460051432385704753352, −1.09513157184347448438907192422, −0.859629915268150574693685825194, 0.859629915268150574693685825194, 1.09513157184347448438907192422, 2.24773209460051432385704753352, 3.01000527337709967171374865417, 3.39567998387535460812683799868, 3.46764598695459551704989318974, 4.35336587019880043428892678064, 4.79260310657974610597233093823, 5.43690883218433792305781071075, 5.55270838919019823822788404583, 6.50250727412608230979435762145, 6.69205073326629599870617058586, 7.08070584991519485848520445672, 7.16052590256910063539794378756, 7.77307587699736122115391753624, 8.312700185860270809450382307879, 9.064455935931297151545476565667, 9.338297278609692927807365341382, 9.452082351369450189442281696885, 9.661165550558986193511889780429

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