Properties

Label 4-30e4-1.1-c1e2-0-13
Degree $4$
Conductor $810000$
Sign $-1$
Analytic cond. $51.6463$
Root an. cond. $2.68077$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 4·13-s + 4·16-s − 4·37-s + 8·49-s + 8·52-s + 16·61-s − 8·64-s − 28·73-s + 20·97-s − 8·109-s + 2·121-s + ⋯
L(s)  = 1  − 4-s − 1.10·13-s + 16-s − 0.657·37-s + 8/7·49-s + 1.10·52-s + 2.04·61-s − 64-s − 3.27·73-s + 2.03·97-s − 0.766·109-s + 2/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(810000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(51.6463\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 810000,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.23.a_bo
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.31.a_bi
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.43.a_adc
47$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \) 2.47.a_ace
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.59.a_w
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.67.a_adc
71$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \) 2.71.a_eo
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \) 2.79.a_afe
83$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \) 2.83.a_ei
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−8.018766790322937442875022215711, −7.60950641627000553986024638811, −7.15281843736937989748818932550, −6.85070007417621569558775442462, −6.06624505516661075058257639643, −5.70544893666969580924943344109, −5.22044744827599394709470298512, −4.79152829469716481173322625979, −4.33130484266058282644341428268, −3.84140494042198101677883902959, −3.25355294631468824600261464826, −2.63413497199860946526293752096, −1.95201361336606951405347226727, −1.00245446974755839368476404741, 0, 1.00245446974755839368476404741, 1.95201361336606951405347226727, 2.63413497199860946526293752096, 3.25355294631468824600261464826, 3.84140494042198101677883902959, 4.33130484266058282644341428268, 4.79152829469716481173322625979, 5.22044744827599394709470298512, 5.70544893666969580924943344109, 6.06624505516661075058257639643, 6.85070007417621569558775442462, 7.15281843736937989748818932550, 7.60950641627000553986024638811, 8.018766790322937442875022215711

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