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Properties

Label 4-1160e2-1.1-c1e2-0-3
Degree $4$
Conductor $1345600$
Sign $1$
Analytic cond. $85.7966$
Root an. cond. $3.04345$
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  + 2·5-s + 2·9-s − 4·13-s + 16·23-s + 3·25-s − 10·29-s + 4·45-s − 14·49-s + 12·53-s + 8·59-s − 8·65-s + 8·67-s − 32·71-s − 5·81-s + 24·83-s + 16·103-s − 24·107-s + 28·109-s + 32·115-s − 8·117-s + 18·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 20·145-s + ⋯
L(s)  = 1  + 0.894·5-s + 2/3·9-s − 1.10·13-s + 3.33·23-s + 3/5·25-s − 1.85·29-s + 0.596·45-s − 2·49-s + 1.64·53-s + 1.04·59-s − 0.992·65-s + 0.977·67-s − 3.79·71-s − 5/9·81-s + 2.63·83-s + 1.57·103-s − 2.32·107-s + 2.68·109-s + 2.98·115-s − 0.739·117-s + 1.63·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.66·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1345600\)    =    \(2^{6} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(85.7966\)
Root analytic conductor: \(3.04345\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1345600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.666184903\)
\(L(\frac12)\) \(\approx\) \(2.666184903\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
29$C_2$ \( 1 + 10 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.23.aq_eg
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.31.a_bm
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.43.a_ade
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.47.a_acg
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.71.bg_pi
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.89.a_agw
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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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

−10.10961794233914650323018370622, −9.482937205001103111761520516085, −9.267695191604174493915302178110, −8.756663769467700445634901809182, −8.709759186685909741113366671102, −7.64281195545062899993026431776, −7.50968928540286871803240472478, −7.18417922295536673306886391318, −6.61185872037464122579958467344, −6.42087341756144188392168238318, −5.58365689991675512377045900017, −5.30185797510899302861480321254, −4.96650299545985705378912087565, −4.49385745654466875698719305336, −3.86871290761964916403115863692, −3.13127304483458464946855247768, −2.84718984244571396597503919941, −2.06881748835706552647444862013, −1.57548947194182609344692708204, −0.73342446702991114092171969869, 0.73342446702991114092171969869, 1.57548947194182609344692708204, 2.06881748835706552647444862013, 2.84718984244571396597503919941, 3.13127304483458464946855247768, 3.86871290761964916403115863692, 4.49385745654466875698719305336, 4.96650299545985705378912087565, 5.30185797510899302861480321254, 5.58365689991675512377045900017, 6.42087341756144188392168238318, 6.61185872037464122579958467344, 7.18417922295536673306886391318, 7.50968928540286871803240472478, 7.64281195545062899993026431776, 8.709759186685909741113366671102, 8.756663769467700445634901809182, 9.267695191604174493915302178110, 9.482937205001103111761520516085, 10.10961794233914650323018370622

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