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Properties

Label 4-2e8-1.1-c14e2-0-0
Degree $4$
Conductor $256$
Sign $1$
Analytic cond. $395.716$
Root an. cond. $4.46011$
Motivic weight $14$
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  + 2.15e5·5-s + 1.74e6·9-s + 1.18e8·13-s − 8.97e7·17-s + 2.26e10·25-s + 6.10e10·29-s − 2.51e11·37-s − 8.28e10·41-s + 3.76e11·45-s − 7.74e11·49-s + 9.46e11·53-s − 1.17e11·61-s + 2.55e13·65-s + 2.92e13·73-s − 1.98e13·81-s − 1.93e13·85-s + 7.17e12·89-s + 1.30e14·97-s − 9.46e12·101-s − 2.79e14·109-s + 3.70e14·113-s + 2.06e14·117-s + 3.28e14·121-s + 1.06e15·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 2.76·5-s + 0.364·9-s + 1.88·13-s − 0.218·17-s + 3.71·25-s + 3.53·29-s − 2.65·37-s − 0.425·41-s + 1.00·45-s − 1.14·49-s + 0.805·53-s − 0.0372·61-s + 5.21·65-s + 2.65·73-s − 0.867·81-s − 0.604·85-s + 0.162·89-s + 1.61·97-s − 0.0882·101-s − 1.52·109-s + 1.57·113-s + 0.688·117-s + 0.864·121-s + 2.24·125-s + 9.76·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+7)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(395.716\)
Root analytic conductor: \(4.46011\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 256,\ (\ :7, 7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(6.183644671\)
\(L(\frac12)\) \(\approx\) \(6.183644671\)
\(L(8)\) 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 \( 1 \)
good3$C_2^2$ \( 1 - 193762 p^{2} T^{2} + p^{28} T^{4} \)
5$C_2$ \( ( 1 - 4314 p^{2} T + p^{14} T^{2} )^{2} \)
7$C_2^2$ \( 1 + 110706500146 p T^{2} + p^{28} T^{4} \)
11$C_2^2$ \( 1 - 2712921553442 p^{2} T^{2} + p^{28} T^{4} \)
13$C_2$ \( ( 1 - 4559650 p T + p^{14} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 44896350 T + p^{14} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 862350223502568242 T^{2} + p^{28} T^{4} \)
23$C_2^2$ \( 1 - 21306379874939894498 T^{2} + p^{28} T^{4} \)
29$C_2$ \( ( 1 - 30520168602 T + p^{14} T^{2} )^{2} \)
31$C_2^2$ \( 1 - \)\(31\!\cdots\!42\)\( T^{2} + p^{28} T^{4} \)
37$C_2$ \( ( 1 + 125971660150 T + p^{14} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 41444675982 T + p^{14} T^{2} )^{2} \)
43$C_2^2$ \( 1 - \)\(13\!\cdots\!18\)\( T^{2} + p^{28} T^{4} \)
47$C_2^2$ \( 1 - \)\(51\!\cdots\!58\)\( T^{2} + p^{28} T^{4} \)
53$C_2$ \( ( 1 - 473374435050 T + p^{14} T^{2} )^{2} \)
59$C_2^2$ \( 1 - \)\(23\!\cdots\!22\)\( T^{2} + p^{28} T^{4} \)
61$C_2$ \( ( 1 + 58526672422 T + p^{14} T^{2} )^{2} \)
67$C_2^2$ \( 1 - \)\(56\!\cdots\!58\)\( T^{2} + p^{28} T^{4} \)
71$C_2^2$ \( 1 + \)\(80\!\cdots\!38\)\( T^{2} + p^{28} T^{4} \)
73$C_2$ \( ( 1 - 14644904168050 T + p^{14} T^{2} )^{2} \)
79$C_2^2$ \( 1 - \)\(66\!\cdots\!62\)\( T^{2} + p^{28} T^{4} \)
83$C_2^2$ \( 1 - \)\(86\!\cdots\!38\)\( T^{2} + p^{28} T^{4} \)
89$C_2$ \( ( 1 - 3585960554322 T + p^{14} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 65197469070850 T + p^{14} T^{2} )^{2} \)
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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

−15.92163319199249814649902996344, −15.53508638071805642537672524827, −14.05635576943637861723533342956, −14.02853446449504548523313315972, −13.50530981485950887714662155355, −12.86967146733674543722078485816, −11.99911416829980436921331354013, −10.76743293064391824892197105426, −10.28298471431699190060176507117, −9.809904113805586099616649339723, −8.860837924954879344064903426070, −8.423837012272146087866772988024, −6.53256606930846835819564950845, −6.52727457433197210817288100159, −5.57922377786305722307423218722, −4.83610788376593993730034581490, −3.39969698437429782264561574663, −2.33982814864191627718871810810, −1.58261540363514476121516155053, −1.01116116653815490583030109822, 1.01116116653815490583030109822, 1.58261540363514476121516155053, 2.33982814864191627718871810810, 3.39969698437429782264561574663, 4.83610788376593993730034581490, 5.57922377786305722307423218722, 6.52727457433197210817288100159, 6.53256606930846835819564950845, 8.423837012272146087866772988024, 8.860837924954879344064903426070, 9.809904113805586099616649339723, 10.28298471431699190060176507117, 10.76743293064391824892197105426, 11.99911416829980436921331354013, 12.86967146733674543722078485816, 13.50530981485950887714662155355, 14.02853446449504548523313315972, 14.05635576943637861723533342956, 15.53508638071805642537672524827, 15.92163319199249814649902996344

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