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Properties

Label 4-2790e2-1.1-c1e2-0-10
Degree $4$
Conductor $7784100$
Sign $1$
Analytic cond. $496.320$
Root an. cond. $4.71998$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 4·5-s + 16-s + 8·19-s + 4·20-s + 11·25-s − 20·29-s − 2·31-s − 4·41-s + 10·49-s − 12·59-s − 16·61-s − 64-s + 16·71-s − 8·76-s − 24·79-s − 4·80-s − 28·89-s − 32·95-s − 11·100-s − 4·109-s + 20·116-s − 22·121-s + 2·124-s − 24·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s + 1/4·16-s + 1.83·19-s + 0.894·20-s + 11/5·25-s − 3.71·29-s − 0.359·31-s − 0.624·41-s + 10/7·49-s − 1.56·59-s − 2.04·61-s − 1/8·64-s + 1.89·71-s − 0.917·76-s − 2.70·79-s − 0.447·80-s − 2.96·89-s − 3.28·95-s − 1.09·100-s − 0.383·109-s + 1.85·116-s − 2·121-s + 0.179·124-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7784100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(496.320\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7784100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.23.a_abq
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.29.u_gc
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.47.a_ada
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.79.y_lq
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.89.bc_ok
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \) 2.97.a_agw
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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

−8.505560084999219944457161997324, −8.267724809024335933901852634956, −7.73747138244806281611040281017, −7.45462785741029316642214847197, −7.21812724275959140155326978769, −7.14595469795646590654050690214, −6.25013352262722179049166927489, −5.84807965935127683751887173870, −5.46569793647563722686364164625, −5.03326406684402643340352897993, −4.75422847809285521609863028506, −3.93781936090595901129745507154, −3.92401283534811723200182494521, −3.57520941155622051122120816839, −3.00412827139072834587607445520, −2.58135226763692579673076694606, −1.56068468832047221486785458455, −1.22639578079610883399352430841, 0, 0, 1.22639578079610883399352430841, 1.56068468832047221486785458455, 2.58135226763692579673076694606, 3.00412827139072834587607445520, 3.57520941155622051122120816839, 3.92401283534811723200182494521, 3.93781936090595901129745507154, 4.75422847809285521609863028506, 5.03326406684402643340352897993, 5.46569793647563722686364164625, 5.84807965935127683751887173870, 6.25013352262722179049166927489, 7.14595469795646590654050690214, 7.21812724275959140155326978769, 7.45462785741029316642214847197, 7.73747138244806281611040281017, 8.267724809024335933901852634956, 8.505560084999219944457161997324

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