Properties

Label 4-1152e2-1.1-c1e2-0-0
Degree $4$
Conductor $1327104$
Sign $1$
Analytic cond. $84.6173$
Root an. cond. $3.03294$
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  − 16·17-s + 6·25-s − 16·41-s − 14·49-s − 12·73-s − 32·89-s + 36·97-s + 32·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 3.88·17-s + 6/5·25-s − 2.49·41-s − 2·49-s − 1.40·73-s − 3.39·89-s + 3.65·97-s + 3.01·113-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1327104\)    =    \(2^{14} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(84.6173\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1152} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1327104,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9355942856\)
\(L(\frac12)\) \(\approx\) \(0.9355942856\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−10.26004222360917259799611803331, −9.526389366691702526219244669729, −9.038896818081904061207915429151, −8.793481022831512363938211169209, −8.392240848196823240102642816951, −8.263873305908931416619055964499, −7.31300450818656449038789296103, −6.91615205120311970156970765799, −6.89321737430630301621976437192, −6.13629272810746063422679438787, −6.10686217759843019636466558994, −5.02848065005896126532311505996, −4.86225409412988049317034377156, −4.46573501861109004967213060939, −3.98969828388277430097709521455, −3.18575233577633167701919505878, −2.85131357734068010185036645810, −1.92655968857620225165092396287, −1.82404577004519249888526111108, −0.39953889036226379155252783473, 0.39953889036226379155252783473, 1.82404577004519249888526111108, 1.92655968857620225165092396287, 2.85131357734068010185036645810, 3.18575233577633167701919505878, 3.98969828388277430097709521455, 4.46573501861109004967213060939, 4.86225409412988049317034377156, 5.02848065005896126532311505996, 6.10686217759843019636466558994, 6.13629272810746063422679438787, 6.89321737430630301621976437192, 6.91615205120311970156970765799, 7.31300450818656449038789296103, 8.263873305908931416619055964499, 8.392240848196823240102642816951, 8.793481022831512363938211169209, 9.038896818081904061207915429151, 9.526389366691702526219244669729, 10.26004222360917259799611803331

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