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Properties

Label 4-930e2-1.1-c1e2-0-12
Degree $4$
Conductor $864900$
Sign $1$
Analytic cond. $55.1467$
Root an. cond. $2.72508$
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  − 4-s + 4·5-s − 9-s + 16-s + 8·19-s − 4·20-s + 11·25-s + 20·29-s − 2·31-s + 36-s + 4·41-s − 4·45-s + 10·49-s + 12·59-s − 16·61-s − 64-s − 16·71-s − 8·76-s − 24·79-s + 4·80-s + 81-s + 28·89-s + 32·95-s − 11·100-s − 4·109-s − 20·116-s − 22·121-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s − 1/3·9-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 + 1/6·36-s + 0.624·41-s − 0.596·45-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 + 1/9·81-s + 2.96·89-s + 3.28·95-s − 1.09·100-s − 0.383·109-s − 1.85·116-s − 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(864900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(55.1467\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.055100252\)
\(L(\frac12)\) \(\approx\) \(3.055100252\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
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.au_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.ae_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.am_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.q_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.abc_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

−10.19162207819299726554368610715, −10.00047568405200539228697810234, −9.274767585490906418972086826201, −9.237516804256183078672722751031, −8.569801162487930724931030941037, −8.537812777193137333475258810287, −7.60075428088826133685036102694, −7.45917870493747497762657713033, −6.63412707880630250998924878427, −6.47160133621760690562630147376, −5.72811720796610628786290294047, −5.70918964423839515412498593212, −4.97314598266374029261205774366, −4.75980100681370702151201742715, −4.11181043515319211873691267605, −3.21238279759276996273909879409, −2.81595747864006454330835131972, −2.41471440131578098084008800367, −1.36574474267146578414953301696, −0.967578865639452462690145627241, 0.967578865639452462690145627241, 1.36574474267146578414953301696, 2.41471440131578098084008800367, 2.81595747864006454330835131972, 3.21238279759276996273909879409, 4.11181043515319211873691267605, 4.75980100681370702151201742715, 4.97314598266374029261205774366, 5.70918964423839515412498593212, 5.72811720796610628786290294047, 6.47160133621760690562630147376, 6.63412707880630250998924878427, 7.45917870493747497762657713033, 7.60075428088826133685036102694, 8.537812777193137333475258810287, 8.569801162487930724931030941037, 9.237516804256183078672722751031, 9.274767585490906418972086826201, 10.00047568405200539228697810234, 10.19162207819299726554368610715

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